Abstract
Given \(d,s \in {\mathbb {N}}\), a finite set \(A \subseteq {\mathbb {Z}}\) and polynomials \(\varphi _1, \dots , \varphi _{s} \in {\mathbb {Z}}[x]\) such that \(1 \le \deg \varphi _i \le d\) for every \(1 \le i \le s\), we prove that
for some \(\eta _s \gg _{d} \log s / \log \log s\). Moreover if \(\varphi _i(0) \ne 0\) for every \(1 \le i \le s\), then
These generalise and strengthen previous results of Bourgain–Chang, Pálvölgyi–Zhelezov and Hanson–Roche-Newton–Zhelezov. We derive these estimates by proving the corresponding low-energy decompositions. The latter furnish further applications to various problems of a sum-product flavour, including questions concerning large additive and multiplicative Sidon sets in arbitrary sets of integers.
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1 Introduction
A central theme in arithmetic combinatorics is the interplay of addition and multiplication over arbitrary sets of integers. In particular, a fundamental phenomenon in the area, which is exhibited and quantified in a rich collection of results, suggests that additive and multiplicative structures in finite sets of integers do not coincide with each other. Thus, given a natural number s and a finite set \(A \subseteq {\mathbb {R}}\), we define the s-fold sumset sA and the s-fold product set \(A^{(s)}\) of A to be
respectively. These can be seen to be standard measures of arithmetic structure, since whenever A is an arithmetic progression, then \(|A| \le |sA| \ll _{s} |A|\), while whenever A is a geometric progression, we have that \(|A| \le |A^{(s)}| \ll _{s} |A|\). Quantifying the aforementioned philosophy of the lack of coexistence between additive and multiplicative structure, Erdős and Szemerédi [6] proposed the following conjecture.
Conjecture 1.1
For any \(s \in {\mathbb {N}}\) and \(\varepsilon >0\) and finite set \(A \subseteq {\mathbb {Z}}\), we have
While this conjecture remains wide open even in the case when \(s=2\), there have been some breakthrough results in this direction, including the beautiful work of Bourgain–Chang [2] that delivers the bound
in the setting of Conjecture 1.1. We refer to this type of an estimate as exhibiting unbounded expansion since the exponent \((\log s)^{1/4} \rightarrow \infty \) as \(s \rightarrow \infty \). Furthermore, this has since been quantitatively improved only once, wherein the exponent \((\log s)^{1/4}\) has been upgraded to \((\log s)^{1 - o(1)}\) by Pálvölgyi–Zhelezov [16].
The sum-product phenomenon has expanded vastly since the work of Erdős and Szemerédi [6], and now encompasses a variety of results which highlight an incongruence between many different types of arithmetic structures. Despite this, there has only been one other result which allows for unbounded expansion, which is due to Hanson, Roche-Newton and Zhelezov [10]. In particular, they showed that for any finite set A of integers and for any \(u \in {\mathbb {Z}} {\setminus } \{0\}\) and \(s\in {\mathbb {N}}\), one has
Recently, an inquiry into a stronger version of sum-product type results was put forth by Balog–Wooley [1]. In order to present this, we first present some definitions, and thus, given \(\varvec{\varphi } = (\varphi _1, \dots , \varphi _{2\,s}) \in ({\mathbb {Z}}[x])^{2\,s}\), we define the mixed energies
and
When \(\varphi _1(x) = \dots = \varphi _{2s}(x) = x\) for every \(x \in {\mathbb {Z}}\), we write \(E_{s}(A) = E_{s, \varvec{\varphi }}(A)\) and \(M_{s}(A) = M_{s, \varvec{\varphi }}(A)\). A standard application of the Cauchy–Schwarz inequality then gives us that
for every \(s \in {\mathbb {N}}\) and every finite \(A \subseteq {\mathbb {Z}}\). Noting this along with Conjecture 1.1, it is natural to expect that for every finite \(A \subseteq {\mathbb {Z}}\), one may write \(A = B \cup C\), with B, C disjoint, such that
for some \(c_s>0\). This was termed by Balog and Wooley as a low-energy decomposition, who proved the first such result with \(c_s < 2/33\). In view of Conjecture 1.1, they further speculated that one should be able to take \(c_s \rightarrow \infty \) as \(s \rightarrow \infty \). Moreover, while many works have subsequently improved admissible exponents in (1.4), till recently, it was still not known whether one may choose \(c_s \rightarrow \infty \) as \(s \rightarrow \infty \). In our recent work [14], this speculation was confirmed quantitatively with [14, Corollary 1.3] delivering the bound \(c_s \gg (\log \log s)^{1/2 - o(1)}\). The latter was further quantitatively improved by Shkredov [25] to \(c_s \gg (\log s)^{1/2 - o(1)}\), and it is widely believed that this type of method limits to delivering exponents of the shape \(c_s \ll \log s/ \log \log s\), due to examples of the form [16, Proposition 1.5].
Our main aim of this paper is to generalise all of the above sum-product results for additive and multiplicative equations over polynomials, and achieve the speculated exponent of \(\log s/\log \log s\) in each of these problems. This is recorded in our main result below.
Theorem 1.2
For any \(d,s \in {\mathbb {N}}\), there exists some \(\eta _s \gg _{d} \log s / \log \log s\) such that the following holds true. Any finite set \(A \subseteq {\mathbb {Q}}\) may be written as \(A = B \cup C\), for disjoint sets B, C, such that for any \(\varvec{\varphi } \in ({\mathbb {Z}}[x])^{2\,s}\) satisfying \(1 \le \deg \varphi _1, \dots , \deg \varphi _{2\,s} \le d\), we have
Moreover, if \(\varphi _i(0) \ne 0\) for each \(1 \le i \le 2s\), then
A first remark about Theorem 1.2 is that it automatically recovers any sumset-product set type estimate of the form (1.1). In particular, writing
for any finite sets \(X, Y\subseteq {\mathbb {R}},\) we see that since \(\max \{|B|, |C|\} \ge |A|/2\) in the conclusion of Theorem 1.2, we may apply Cauchy–Schwarz inequality, in a manner akin to (1.3), to obtain the following estimate on sumsets and product sets.
Corollary 1.3
For any \(d,s \in {\mathbb {N}}\) and any \(\varphi _1, \dots , \varphi _s \in {\mathbb {Q}}[x]\) such that \(1 \le \deg \varphi _i \le d\) for each \(1 \le i \le s\) and for any finite set \(A \subseteq {\mathbb {Q}}\), we have that
for some \(\eta _s \gg _{d} \log s / \log \log s\). Moreover if \(\varphi _i(0) \ne 0\) for every \(1 \le i \le s\), then
Note that upon setting \(\varphi _1, \dots , \varphi _s\) to be the same appropriately chosen linear polynomial in Corollary 1.3, we are able to automatically recover the best known bound for Conjecture 1.1 that is presented in [16] as well as quantitatively strengthen the main result of [10]. In fact, Corollary 1.3 presents the first set of bounds wherein one may choose a variety of different polynomials \(\varphi _1, \dots , \varphi _s\) with unlike but bounded degrees and still obtain unbounded expansion. We further note that these so-called low-energy decompositions are much stronger than the corresponding sumset-product set type estimates. For instance, bounds of the form (1.6) are new even in the situation when, say, \(\varphi _1(x) = \dots = \varphi _{2s}(x) = c x + d\) for every \(x \in {\mathbb {Z}}\), for some fixed \(c,d \in {\mathbb {Z}}{\setminus }\{0\}\), while the corresponding sumset-product set type estimate was already proven, albeit with a quantitatively weaker exponent, in [10]. Moreover, Theorem 1.2 allows us to makes progress on some other problems with a sum-product flavour, which themselves generalise estimates of the form (1.1). We describe one such result below.
Thus, given \(\varphi \in {\mathbb {Z}}[x]\), we denote a finite set \(X \subseteq {\mathbb {Z}}\) to be a \(B_{s, {\varphi }}^+[1]\) set if for every \(n \in {\mathbb {Z}}\), there is at most one distinct solution to the equation
with \(x_1, \dots , x_s \in X\). Here, we consider two such solutions to be the same if they differ only in the ordering of the summands. Similarly, we denote X to be a \(B_{s, \varphi }^{\times }[1]\) set if for every \(n \in {\mathbb {Z}}\), there is at most one distinct solution to the equation
with \(x_1, \dots , x_s \in X\). When \({\varphi }\) equals the identity map, then a \(B_{s, {\varphi }}^+[1]\) set is known as an additive Sidon set, which we will denote as a \(B_{s}^+[1]\) set. We define a \(B_{s}^{\times }[1]\) set in an analogous fashion, and we refer to this as a multiplicative Sidon set. These objects play a central role in the field of combinatorial number theory, and there has been a rich line of work investigating their various properties, see [5] and the references therein. While the main such inquiry often surrounds the size of the largest such set which is contained in an ambient set of integers, such as, say \(\{1, 2, \dots , N\}\), this has recently been generalised to the setting of finding \(B_{s}^+[1]\) and \(B_s^{\times }[1]\) sets in arbitrary finite sets of integers, with the hope of having the size of at least one of them being much closer to the size of the ambient set, see [11, 18]. For instance, it is known that any finite set \(A \subseteq {\mathbb {Z}}\) contains a \(B_{s}^{+}[1]\) set of size at least \(c_s' |A|^{1/s}\) for some \(c_s' >0\) (see [12, 22]), and this is sharp, up to multiplicative constants, by setting \(A = \{1, \dots , N\}\). On the other hand, \(\{1, \dots , N\}\) contains large \(B_{s}^{\times }[1]\) sets of size at least \(N(\log N)^{-1}(1 - o(1))\), consider, for example, the set of primes up to N.
It was shown in [11] that results akin to Theorem 1.2 may be employed to make progress on such problems, and in this paper, we record some further improvements along this direction by proving the following result.
Theorem 1.4
For any \(d, s \in {\mathbb {N}}\), there exists a parameter \(\delta _s \gg _{d} \log s/ \log \log s\) such that the following holds true. Given any finite set \(A \subseteq {\mathbb {Z}}\) and any \({\varphi } \in {\mathbb {Z}}[x]\) such that \(\deg \varphi = d\), the largest \(B_{s, {\varphi }}^+[1]\) subset X of A and the largest \(B_{s}^{\times }[1]\) subset Y of A satisfy
Moreover if \(\varphi (0) \ne 0\), then writing \(X'\) to be the largest \(B_{s, \varphi }^{\times }[1]\) subset of A, we have
The first such result was proven in [11], where, amongst other estimates, it was shown that any finite set A of integers contains either a \(B_{s}^{+}[1]\) set or a \(B_{s}^{\times }[1]\) set of size at least \(|A|^{\delta _s/s}\), with \(\delta _s \gg (\log \log s)^{1/2 - o(1)}\). This was subsequently improved by Shkredov [25] who gave the bound \(\delta _s \gg (\log s)^{1/2 - o(1)}\). Theorem 1.4 not only quantitatively improves this to \(\delta _s \gg (\log s)^{1 - o(1)}\), but it also generalises this to estimates on \(B_{s, {\varphi }}^{+}[1]\) and \(B_{s, {\varphi }}^{\times }[1]\) sets, for suitable choices of \({\varphi } \in {\mathbb {Z}}[x]\). Moreover, this can be seen as an alternate generalisation of the sum-product phenomenon, since the first conclusion of Theorem 1.4 implies that
As before, this recovers the best known result towards Conjecture 1.1 for large values of s by simply setting \(\varphi \) to be the identity map.
In consideration of Conjecture 1.1, one may naively expect Theorems 1.2 and 1.4 to hold for every \(\eta _s < s\) and \(\delta _s <s\) respectively, since either of these would imply Conjecture 1.1 in a straightforward manner, but in fact, both of these have been shown to be false in various works. In [1], Balog–Wooley constructed arbitrarily large sets A of integers, such that for every \(B \subseteq A\) with \(|B| \ge |A|/2\), we have that
thus implying that \(\eta _s \le (2s+1)/3\). Similarly, in the case when \(\varphi \) is set to be the identity map, it was noted by Roche-Newton that these sets also gave the bound \(\delta _2 \le 3/2\), thus refuting a question of Klurman–Pohoata [18]. This was then improved by constructions of Green–Peluse (unpublished), Roche-Newton–Warren [21] and Shkredov [24] to \(\delta _2 \le 4/3\). The latter constructions were then generalised in [11, Proposition 1.5] for every \(s \in {\mathbb {N}}\), thus giving the estimate \(\delta _s/s \le 1/2 + 1/(2\,s+2)\) when s is even and \(\delta _s/s \le 1/2 + 1/(2\,s)\) when s is odd.
We now return to the even more general case where s, d are some natural numbers and \(\varphi \in {\mathbb {Z}}[x]\) is some arbitrary polynomial with \(\deg \varphi = d\). In this setting, we are able to utilise ideas from [1] and [11] to record the following upper bounds.
Proposition 1.5
Let d, s, N be natural numbers and let \(\varphi \in ({\mathbb {Z}}[x])^{2\,s}\) satisfy \(\varvec{\varphi } = (\varphi , \dots , \varphi )\) for some \(\varphi \) with \(\deg \varphi = d\). If \(s \ge 10d(d+1)\), then there exists a finite set \(A \subseteq {\mathbb {N}}\) satisfying \(N \ll |A|\) such that for any \(B \subseteq A\) with \(|B| \ge |A|/2\), we have that
Similarly, when s is even, there exists a finite set \(A' \subseteq {\mathbb {N}}\) with \(N \ll |A'|\) such that the largest \(B_{s, {\varphi }}^+[1]\) subset X and the largest \(B_{s}^{\times }[1]\) subset Y of \(A'\) satisfy
This, along with Theorems 1.2 and 1.4, implies that whenever \(s \ge 10d(d+1)\), then
holds true. Thus, there is a large gap between the known upper and lower bounds, and it would be interesting to understand the right order of magnitude for these quantities, see also [11, Question 1.6]. We remark that Proposition 9.2 provides further relations between Theorems 1.2 and 1.4.
Returning to our original theme, we note that a combination of Theorem 1.2 and the Plünnecke–Ruzsa inequality (see Lemma 6.1) implies that whenever a set \(A \subseteq {\mathbb {Z}}\) satisfies \(|A^{(2)}| \le K|A|\) with \(K = |A|^{\eta _s/10\,s}\), then the sumset \(\varphi _1(A) + \cdots + \varphi _s(A)\) exhibits unbounded expansion. But in fact, in this case, we are able to show that such sumsets must be close to being extremally large. This can be deduced from the following more general result. As is usual, we write \(e(\theta ) = e^{2 \pi i \theta }\) for every \(\theta \in {\mathbb {R}}\).
Theorem 1.6
Let \(K \ge 1\) be a real number and let d, s be natural numbers. Moreover, let \(A \subseteq {\mathbb {Z}}\) be a finite set such that \(|A^{(2)}| = K|A|\) and let \(\varphi \in {\mathbb {Z}}[x]\) satisfy \(\deg \varphi = d\) and let \({\mathfrak {a}}: {\mathbb {Z}} \rightarrow [0, \infty )\) be a function. Then, writing \(C = 8 + 12 \log (d^2 + 2) + 6 \log (2\,s)\), we have
When \(\varphi (x) = x\) for every \(x \in {\mathbb {Z}}\), such types of results have been previously referred to as the weak Erdős–Szemerédi Conjecture, which suggests that whenever \(|A^{(2)}| \le K|A|\), then we have \(|2A| \gg _{\varepsilon } |A|^{2 - \varepsilon }/ K^{O(1)}\) for every \(\varepsilon >0\). Proving such an estimate formed a key step in the work of Bourgain–Chang [2], and in fact, Theorem 1.6 improves upon [16, Theorem 1.3] by replacing small powers of |A| with a factor of \((\log |A|)^{O(1)}\).
Denoting the mean value on the left hand side of (1.7) to be \(E_{s, {\mathfrak {a}}, \varphi }(A)\), we may apply orthogonality to see that
where the inequality follows from counting the diagonal solutions \(a_{i} = a_{i+s}\) for all \(1 \le i \le s\). On the other hand, given \(\varepsilon >0\), in the case when \(|A^{(2)}| \ll _{s,d} |A|^{ c\varepsilon }|A|\) for some small constant \(c = c(d,s)>0\), we may apply Theorem 1.6 to deduce that
which matches the aforementioned lower bound up to a factor of \(|A|^{\varepsilon }\). Thus, Theorem 1.6 indicates that additive polynomial equations exhibit strongly diagonal type behaviour with respect to multiplicatively structured sets of integers. In fact, our methods can prove a corresponding result for multiplicative polynomial equations, which, in turn, can be employed to furnish a non-linear analogue of a subspace-type theorem, see Theorems 2.1 and 2.2.
We remark that apart from the sum-product conjecture, our results are motivated by another well-known phenomenon which studies growth of sets of the form \(F(A, \dots , A) = \{ F(a_1, \dots , a_s) \ | \ a_1, \dots , a_s \in A\}\), where F is an arbitrary polynomial in s variables. Estimates for \(|F(A, \dots , A)|\), as well as its connection to the size of the product set \(A^{(s)}\), have been widely studied in previous works, see, for instance [3, 19, 20, 26]. In most of such papers, the motivation is to find some \(0<c\le 1\) such that either \(|F(A, \dots , A)|\gg |A|^{1+ c}\) holds for all large, finite subsets A of some ambient set, or F is of a special form, such as, say \(F(x_1, \dots , x_s) = \varphi _1(x_1) + \cdots + \varphi _s(x_s)\) or \(F(x_1, \dots , x_s) = \varphi _1(x_1) \dots \varphi _s(x_s),\) where \(\varphi _1, \dots , \varphi _s\) are univariate polynomials. On the other hand, our results show that over \({\mathbb {Q}}\), if F is of the latter type with suitably chosen \(\varphi _1, \dots , \varphi _s\), then either \(|F(A, \dots , A)|\) or \(|A^{(s)}|\) exhibits unbounded expansion. Thus our results compliment the regimes analysed by these previous works. Moreover, the methods involved in the latter seem to be quite different from the techniques used in our paper.
We now proceed to describe the outline of our paper, along with some of the proof ideas present therein. As previously mentioned, Sect. 2 is dedicated to presenting some further applications of our method. In Sect. 3, we record some properties of the mixed energies \(E_{s, \varvec{\varphi }}(A)\) and \(M_{s, \varvec{\varphi }}(A)\) that we will use throughout our paper. The first main step towards proving Theorem 1.2 is initiated in Sect. 4, which we utilise to prove a generalisation of a result of Chang [4, Proposition 8]. This is the content of Lemma 4.1, which can be interpreted as a decoupling type inequality. For instance, in the additive case, suppose that we have natural numbers d, r, s, a prime number p, a polynomial \(\varphi \in {\mathbb {Z}}[x]\) with \(\deg \varphi = d\), and finite sets \(A_0, A_1, \dots , A_r\) of natural numbers such that for every \(1 \le i \le r\), we have \(A_i = \{ a \in A_0 \ | \ \nu _p(a) = n_i \}\) for some unique \(n_i \in {\mathbb {N}} \cup \{0\}\), where \(\nu _p(n)\) denotes the largest exponent \(m \in {\mathbb {N}} \cup \{0\}\) such that \(p^m\) divides n. Writing
where \({\mathfrak {a}}: {\mathbb {N}} \rightarrow [0, \infty )\) is some function, we are interested in proving estimates of the form
that is, we want to exhibit square-root cancellation in moments of these exponential sums. This may then be iterated to obtain estimates on \(E_{s, {\mathfrak {a}}, \varphi }(A)\) in terms of the \(l^2\)-norm of \({\mathfrak {a}}\) and the so-called query-complexity q(A) of A, see Lemma 4.2 for more details.
We now proceed to Sect. 5, where we analyse the multiplicative analogue of this phenomenon. This ends up being harder to deal with, and in particular, we have to first study some auxilliary mean values of the form
and prove decoupling type inequalities for such quantities. Iterating these estimates, as in Sect. 4, leads to suitable bounds for \(J_{s, {\mathfrak {a}}, \varphi }(A)\). Next, we prove a multiplicative variant of an averaging argument from analytic number theory, which allows us to discern bounds on \(M_{s, {\mathfrak {a}}, \varphi }(A)\) from estimates for \(J_{s, {\mathfrak {a}}, \varphi }(A)\) by incurring a further factor of \(|A^{(s)}/A^{(s)}|\), see Lemmata 5.1 and 5.4.
At this stage of the proof, we collect various inverse theorems from arithmetic combinatorics in Sect. 6, the first of these arising from our work on a variant of the s-fold Balog–Szemerédi–Gowers theorem in [14]. This, along with a dyadic pigeonholing trick, allows us to deduce that whenever A satisfies \(M_{s}(A) \ge |A|^{2s- k}\), for suitable values of s, k, then A has a large intersection with a set \(U'\) satisfying \(|U'| \ll |A|^k\), such that the many-fold product sets of \(U'\) expand slowly, see Theorem 6.2 for more details. The other inverse theorem that we are interested in emanates from the circle of ideas recorded in [16], and it implies that given two finite sets \(A, X \subseteq {\mathbb {N}}\) satisfying the inequality \(|A \cdot X \cdot X| \le K |X|\), there must exist a large subset B of A with \(q(B) \le \log K\). Roughly speaking, this can be seen as a Freiman type structure theorem for sets with small asymmetric product sets, since query-complexity itself may be interpreted as a skewed version of some notion of multiplicative dimension. The two aforementioned inverse theorems combine naturally to imply that any set \(A \subseteq {\mathbb {Z}}\) with a large multiplicative energy has a large subset B with a small query complexity. In Sect. 7 we apply this idea iteratively, along with the results recorded in Sects. 3–6, to yield the proof of Theorem 1.2. We utilise Sect. 8 to prove Theorems 1.6 and 2.1, and finally, in Sect. 9, we provide the proofs of Theorem 1.4 and Proposition 1.5.
Notation. In this paper, we use Vinogradov notation, that is, we write \(X \gg _{z} Y\), or equivalently \(Y \ll _{z} X\), to mean \(X \ge C_{z} |Y|\) where C is some positive constant depending on the parameter z. We use \(e(\theta )\) to denote \(e^{2\pi i \theta }\) for every \(\theta \in {\mathbb {R}}\). Moreover, for every natural number \(k \ge 2\) and for every non-empty, finite set Z, we use |Z| to denote the cardinality of Z, we write \(Z^k = \{ (z_1, \dots , z_k) \ | \ z_1, \dots , z_k \in Z\}\) and we use boldface to denote vectors \(\varvec{z} = (z_1, z_2, \dots , z_k) \in Z^k\). All our logarithms will be with respect to base 2.
2 Further applications
As mentioned in the previous section, we are able to prove various multiplicative analogues of Theorem 1.6. In order to state these, we first define a generalisation of \(M_{s, \varvec{\varphi }}(A)\) and \(J_{s, {\mathfrak {a}}, \varphi }(A)\), and so, for every \(\varvec{\varphi } \in {\mathbb {Q}}[x]^{2s}\), every \({\mathfrak {a}}: {\mathbb {R}} \rightarrow [0, \infty )\) and every finite set \(A \subseteq {\mathbb {R}}\), we define
and
Moreover, for any \(\varphi \in {\mathbb {R}}[x]\), we write \({\mathcal {Z}}_{\varphi } = \{ x \in {\mathbb {R}} \ | \ \varphi (x) = 0\}\). Similarly, for any \(\varvec{\varphi } = (\varphi _1, \dots , \varphi _{2\,s}) \in ({\mathbb {R}}[x])^{2\,s}\), we denote \({\mathcal {Z}}_{\varvec{\varphi }} = {\mathcal {Z}}_{\varphi _1} \cup \dots \cup {\mathcal {Z}}_{\varphi _{2\,s}}\). Using methods related to the proofs of Theorems 1.2 and 1.6, we are able to derive the following upper bounds for \(J_{s, {\mathfrak {a}}, \varvec{\varphi }}(A)\) and \(M_{s, {\mathfrak {a}}, \varvec{\varphi }}(A)\), whenever \(A \cdot A\) is small.
Theorem 2.1
Let \(K \ge 1\) be a real number and let d, s be natural numbers. Moreover, let \(\varvec{\varphi } \in {\mathbb {Q}}[x]^{2\,s}\) satisfy \(1 \le \deg \varphi _i \le d\) for each \(1 \le i \le 2s\), let A be a finite subset of \({\mathbb {Q}} {\setminus }( {\mathcal {Z}}_{\varvec{\varphi }}\cup \{0\})\) such that \(|A \cdot A| = K|A|\), and let \({\mathfrak {a}}: {\mathbb {N}} \rightarrow [0, \infty )\) be a function. Then
where \(C = 10 + 48\log (d+3) + 6 \log (2s)\).
Here, the condition \(A \subseteq {\mathbb {Q}}{\setminus }( {\mathcal {Z}}_{\varvec{\varphi }}\cup \{0\})\) seems to be necessary. In order to see this, we may choose \(\varphi \in {\mathbb {Q}}[x]\) to be some linear polynomial, \(\varvec{\varphi } = (\varphi , \dots , \varphi )\) and \(A = \{2,4,\dots , 2^N\} \cup \{x\}\) for some \(x \in {\mathcal {Z}}_{\varphi }\). In this case, since \(\varphi (a_1) \dots \varphi (a_{s-1}) \varphi (x) = \varphi (a_{s+1}) \dots \varphi (a_{2\,s-1}) \varphi (x) = 0\) for any \(a_1, \dots , a_{s-1}, a_{s+1}, \dots , a_{2\,s-1} \in A\), we see that \(J_{s, \varvec{\varphi }}(A) \ge M_{s-1}(A) \gg _s N^{2\,s - 3}\).
We will now use the above result to prove a non-linear analogue of a subspace-type theorem. In particular, it was noted in [10] that a quantitative version of the well-known subspace theorem of Evertse, Schmidt and Schlikewei [7] combined together with Freiman’s lemma [27, Lemma 5.13] implies that for any fixed \(c_1, c_2 \in {\mathbb {C}} {\setminus } \{0\}\) and for any finite subset \(A \subseteq {\mathbb {Q}}\) with \(|A \cdot A| = K|A|\), we have
In fact, a more general result can be proven via these techniques which holds for linear equations with many variables (see [10, Corollary 1.6]), but for simplicity of exposition, we restrict to the two-fold case here. While the above inequality is very effective for small values of K; it may deliver worse than trivial bounds when K is large. For instance, in the case when \(K > c \log |A|\) for some constant \(c > (4\log 2)^{-1}\), the right hand size becomes much larger than the trivial upper bound |A|. The authors of [10] asked whether the above upper bound could be improved to have a polynomial dependence in K, and proved that for any fixed \(c_1, c_2 \in {\mathbb {Q}} {\setminus } \{0\}\) and for any \(\varepsilon >0\) and \(A \subseteq {\mathbb {Q}}\) with \(|A \cdot A| = K|A|\), one has
for some constant \(C_{\varepsilon } >0\). Using Theorem 2.1, we can prove a non-linear analogue of the above result.
Theorem 2.2
Let A be a finite subset of \({\mathbb {Q}}\) such that \(|A \cdot A| \le K|A|\) for some \(K \ge 1\), let \(\varphi \in {\mathbb {Q}}[x]\) have \(\deg \varphi = d \ge 1\) with \(\varphi (0) \ne 0\) and let \(\varepsilon >0\). Then
for some constant \(C = C(d,\varepsilon ) >0\).
Proof
We begin by partitioning \({\mathbb {R}}\) as \({\mathbb {R}} = I_1 \cup \dots \cup I_{r} \cup I_{r+1}\), where \(r \ll _d 1\) and \(I_1, \dots , I_r\) are open intervals such that the derivative \(\varphi '\) is non-zero on each such interval and such that the set \(Z_{\varphi } \cup \{0\} \subseteq I_{r+1}\) with \(|I_{r+1}| \ll _{d} 1\). Let \(A_i = A \cap I_{i}\) for each \(1 \le i \le r+1\). Next, given any two finite sets X, Y, we denote r(X, Y) to be the number of solutions to the equation \(x = \varphi (y)\), with \(x \in X, y \in Y\). With this notation in hand, we note that for any \(1 \le i \le r\), we have that \(r(A_{r+1}, A_i) + r(A_i, A_{r+1}) \ll _d |A_{r+1}| \le |I_{r+1}| \ll _d 1\) since fixing either of x, y in the equation \(x = \varphi (y)\) fixes the other variable up to \(O_d(1)\) choices. Thus, it suffices to upper bound \(r(A_i, A_j)\) for every \(1 \le i, j \le r\). Fixing some \(1 \le i,j \le r\), we define \(S = \{ a \in A_j: \varphi (a) \in A_i\}\), whence, \(r(A_i,A_j) \ll _d |S|\). Since \(S \subseteq A\), we may utilise Theorem 2.1 to deduce that for every \(s \ge 2\), we have
where \(C = 10 + 24 \log (d+3) + 6 \log (2s)\). Applying Cauchy–Schwarz inequality, we get that
On the other hand, since \(\varphi (S) \subseteq A_i \subseteq A\), we may apply the Plünnecke–Ruzsa inequality (see Lemma 6.1) to deduce that
This, together with the preceding expression, implies that
for every \(s \ge 2\). Noting the fact that \(\log |A| \ll _{\varepsilon } |A|^{\varepsilon /4}\), we may choose s to be sufficiently large in terms of \(\varepsilon \), say, \(s = \lceil 10 \epsilon ^{-1}\rceil \) to obtain the desired result. \(\square \)
Theorem 2.2 can be interpreted as a bound on the number of points of \(A \times A\) that lie on the polynomial curve \(y = \varphi (x)\). Naturally, one may employ this result to further prove an incidence estimate for point sets that are of the form \(A \times A\), with \(A \subseteq {\mathbb {Q}}\) satisfying \(|A \cdot A| \le K|A|\) for some parameter \(K \ge 1\), and finite sets of curves of the form \(y = \varphi (x)\), where \(\varphi \in {\mathbb {Q}}[x]\) satisfies \(\varphi (0) \ne 0\), but we do not pursue this here.
3 Properties of mixed energies
We begin this section by recording some notation. For every real number \(\lambda \ne 0\) and every \(\varphi \in {\mathbb {R}}[x]\), we define the polynomial \(\varphi _{\lambda } \in {\mathbb {R}}[x]\) by writing \(\varphi _{\lambda }(x) = \varphi (\lambda x)\) for every \(x \in {\mathbb {R}}\). Furthermore, given \(s \in {\mathbb {N}}\) and \(\varvec{\varphi } \in {\mathbb {R}}[x]^{s}\), we denote \(\varvec{\varphi }_{\lambda } = (\varphi _{1, \lambda }, \dots , \varphi _{s, \lambda })\) and we write \(\lambda \cdot \varvec{\varphi } = (\lambda \varphi _1, \dots , \lambda \varphi _{s})\). It is worth noting that
where for every finite set \(X \subseteq {\mathbb {R}}\) and for every \(\nu \in {\mathbb {R}}\), we denote \(\nu \cdot X = \{ \nu x \ | \ x \in X\}\). Next, given some function \({\mathfrak {a}}: {\mathbb {R}} \rightarrow [0, \infty )\) and some finite sets \(A_1, \dots , A_{2\,s} \subseteq {\mathbb {R}}\), we define
If \(A_1 = \dots = A_{2s} = A\), we denote \(E_{s, {\mathfrak {a}}}(A) = E_{s, {\mathfrak {a}}}(A_1, \dots A_{2\,s})\). Moreover, when the function \({\mathfrak {a}}\) satisfies \({\mathfrak {a}}(x) = 1\) for every \(x \in {\mathbb {R}}\), we suppress the dependence on \({\mathfrak {a}}\), and thus, we write \(E_{s}(A_1, \dots , A_{2\,s}) = E_{s, {\mathfrak {a}}}(A_1, \dots , A_{2\,s})\).
Our first aim in this section is to prove the following generalisation of [14, Lemma 3.2].
Lemma 3.1
Let \(A_1, \dots , A_{2s}\) be a finite sets of real numbers and let \({\mathfrak {a}}: {\mathbb {R}} \rightarrow [0, \infty )\). Then
Moreover, for natural number r and for finite subsets \(A_1, \dots , A_r \subseteq {\mathbb {R}}\), we have
Finally, if \({\mathfrak {a}}(x) = 1\) for every \(x \in {\mathbb {R}}\), then for every \(1 \le l \le s\), we have
Proof
We begin by focusing on proving the first inequality. Firstly, note that this inequality remains invariant under replacing the function \({\mathfrak {a}}\) by \({\mathfrak {a}}/M\), for any \(M > 0\). Moreover, since \(A_1,\dots , A_{2s}\) are finite sets, we may set \(M = \max _{a \in A_1 \cup \dots \cup A_{2\,s}} {\mathfrak {a}}(a) + 100\) in the preceding statement to ensure that \(0 \le {\mathfrak {a}}(a) \le 1\) for any \(a \in A_1 \cup \dots \cup A_{2\,s}\). We further point out that it suffices to prove that for every \(\varepsilon >0\), we have
For the purposes of this proof, we define, for each \((\xi , R) \in {\mathbb {R}}^2\) satisfying \(\xi \ne 0\) and \(R>0\), the quantity \(I(R, \xi ) = \int _{[0,R]} e( \xi \alpha ) d \alpha \). When \(\xi \ne 0\), we see that \(|I(R, \xi )| \ll |\xi |^{-1}\) while \(I(R,0) = R\). We now define, for each \(1 \le i \le 2s\), the exponential sum \(f_i:[0, \infty ) \rightarrow {\mathbb {C}}\) as
and we let \(X = A_1 \cup \dots \cup A_{2s}\). Finally, we write
With this discussion in hand, it is straightforward to note that for each \(R>0\), we have
whence,
where we have used the fact that \(0 \le {\mathfrak {a}}(a) \le 1\) for any \(a \in A_1 \cup \dots \cup A_{2s}\). Similarly, for each \(1 \le i \le 2\,s\), we have that
Amalgamating these expressions with a standard application of Hölder’s inequality gives us
Choosing R to be sufficiently large, say \(R \ge \varepsilon ^{-1} (4\,s^2 |A_1| \dots |A_{2\,s} |)^{4\,s^2} \xi _0^{-1}\), delivers the first inequality stated in our lemma. The second inequality can be swiftly deduced from the first inequality since
The proof of the third inequality follows similarly, wherein, we see that it suffices to show
for each \(\varepsilon >0\). This follows from noting the fact that \(|f_1(\alpha )| \le |A|\) for each \(0 \le \alpha \le R\) along with (3.1), and then choosing R to be some sufficiently large real number. \(\square \)
We now generalise Lemma 3.1 for mixed energies of the form \(J_{s, {\mathfrak {a}}, \varvec{\varphi }}(A)\). Thus, let \(A_1, \dots , A_{2\,s}\) be finite subsets of \({\mathbb {Z}}\), let \(\varvec{\varphi } \in {\mathbb {Z}}[x]^{2s}\) be a vector and let \({\mathfrak {a}}:{\mathbb {N}} \rightarrow [0, \infty )\) be a function supported on \(A_1 \cup \dots \cup A_{2s}\). We define \(J_{s, {\mathfrak {a}}, \varvec{\varphi }}(A_1, \dots , A_{2\,s})\) to be the quantity
that is, a weighted count of the number of solutions to the system of equations
with \(x_i \in A_i\) for each \(1 \le i \le 2s\). We now present our second lemma that we will prove in this section.
Proposition 3.2
Let \(\varvec{\varphi } \in {\mathbb {Q}}[x]^{2s}\) be a vector such that \(\deg \varphi _i \le d\) for every \(1 \le i \le 2s\) and let \(A_1, \dots , A_{2\,s}\) be finite subsets of \({\mathbb {Q}}{\setminus } (\{0\} \cup {\mathcal {Z}}_{\varvec{\varphi }})\), and let \({\mathfrak {a}}: {\mathbb {Q}} \rightarrow [0,\infty )\) be a function. Then we have that
Moreover, if for each \(1 \le j \le 2s\), we have some open interval \(I_j \subseteq {\mathbb {R}}\) such that \(x y >0\) and \(\varphi _i(x) \varphi _i(y) > 0\) for every \(x,y \in I_j\) and \(A_j \subseteq I_j\), then
Proof
We begin by noting that
holds true if and only if we have
for every \({\lambda } \ne 0\). Moreover, note that
Hence, upon dilation by an appropriate natural number \(\lambda \), we may assume that \(\varvec{\varphi }\) is an element of \({\mathbb {Z}}[x]^{2s}\) as well as that \(A_1, \dots , A_{2s}\) are finite subsets of \({\mathbb {Z}}{\setminus } (\{0\} \cup {\mathcal {Z}}_{\varvec{\varphi }})\).
Next, let X be a finite subset of \({\mathbb {N}}^2\). Then, there exists a sufficiently large distinct prime number p such that the map \(\varrho _{X}: X \rightarrow {\mathbb {N}}\), defined as
for all primes \(p_1, \dots , p_r,q_1, \dots , q_t\) and for all non-negative integers \(\alpha _1, \dots , \alpha _r, \beta _1, \dots , \beta _t, r, t\), is bijective onto its image and satisfies the fact that for every \((x_1, y_1), \dots , (x_{2s}, y_{2s}) \in X\), we have
if and only if
Moreover, we can further define the logarithmic map \(\psi _{X}: \varrho _{X}(X) \rightarrow [0, \infty )\) by writing \(\psi _{X}(n) = \log n\) for every \(n \in \varrho _X(X)\). Note that \(\psi _{X}\) is bijective onto its image and satisfies the fact that for every \(z_1, \dots , z_{2\,s} \in \varrho _X(X)\), we have that
With these preliminary manoeuvres finished, we will now proceed to prove our proposition. First, we will show that (3.4) implies (3.3), and then we will prove (3.4). In order to prove the first part, note that for each \(1 \le i \le 2\,s\), we may partition
where \(I_{i,1}, \dots , I_{i,r_i}\) are open, pairwise disjoint intervals such that \(\varphi _i(x) \varphi _i(y)>0\) and \(xy >0\) for all \(x,y \in I_{i,j}\), where \(1 \le j \le r_i\) and \(r_i \le d+2\). Writing \(A_{i,j} = A_{i} \cap I_{i,j}\) for each \(1 \le i \le 2s\) and \(1 \le j \le r_i\), we see that
where we have crucially used the fact that \(A_i = \cup _{1 \le j \le r_i} A_{i, j}\) for each \(1 \le i \le 2s\), which itself follows from the hypothesis that \(A_1, \dots , A_{2s}\) are subsets of \({\mathbb {Q}} {\setminus } (\{0\} \cup {\mathcal {Z}}_{\varvec{\varphi }})\). Combining the preceding expression with (3.4) and the fact that \(r_1, \dots , r_{2s} \le d+2\) then delivers (3.3).
We will now prove (3.4), and so, we define the set \(X_i = \{ ( |a|, |\varphi _i(a)| ): a \in A_i \}\) for each \(1 \le i \le 2\,s\). Note that \(J_{s, {\mathfrak {a}}, \varvec{\varphi }}(A_1, \dots , A_{2\,s})\) is bounded by the number of solutions to the system
with \(a_i \in A_i\) for each \(1 \le i \le 2s\), where each solution is being counted with the weight \({\mathfrak {a}}(a_1) \dots {\mathfrak {a}}(a_{2\,s})\). Consequently, letting \(X = \cup _{1 \le i \le 2\,s} X_i\) and \(\sigma (x) = \psi _{X}(\varrho _{X}(x))\) for every \(x \in X\), we see that
where the last inequality follows from Lemma 3.1. By the definition of \(\sigma \), we see that for each \(1 \le i \le 2s\), the quantity \(E_{s,{\mathfrak {a}}}(\sigma (X_i))\) is equal to the number of solutions to the system
with \(a_1, \dots , a_{2s} \in A_i\), where each such solution is counted with the weight \({\mathfrak {a}}(a_1) \dots {\mathfrak {a}}(a_{2\,s})\). This, in turn, equals \(J_{s, {\mathfrak {a}}, \varphi _i}(A_i)\) since the functions x and \(\varphi _i(x)\) do not change signs as x varies in \(A_i\). Combining this with the preceding discussion, we get that
which is the desired bound. \(\square \)
Our final result in this section allows us to prove various other relations between the above type of mixed energies.
Lemma 3.3
Let d, s, r be natural numbers, let \(\varvec{\varphi } \in ({\mathbb {Q}}[x])^{2\,s}\) satisfy \(\deg \varphi _i \le d\) for each \(1 \le i \le 2\,s\). Then for all finite subsets \(A_1, \dots , A_{r}\) of \({\mathbb {Q}}{\setminus } (\{0\} \cup {\mathcal {Z}}_{\varvec{\varphi }})\) and for every function \({\mathfrak {a}}: {\mathbb {Q}} \rightarrow [0,\infty )\), we have that
For every finite subset A of \({\mathbb {Q}}{\setminus } (\{0\}\cup {\mathcal {Z}}_{\varphi _1})\), for every \(1 \le l < s\) and for the function \({\mathfrak {a}}(x) = 1\) for each \(x \in {\mathbb {R}}\), we have
Proof
As before, (3.5) follows from Proposition 3.2 in a straightforward manner since
We will now outline the proof of (3.6). As in the proof of Proposition 3.2, we see that upon losing a factor of \(O_{s,d}(1)\), it suffices to consider the case when \(A \subseteq I\), where I is some interval such that the functions \(\varphi _1(x)\) and x do not change signs as x varies in I. As before, this allows us to construct a finite set \(Z \subseteq (0, \infty )\) such that \(|Z| = |A|\) and \(J_{t, {\mathfrak {a}}, \varphi _1}(A) = E_{t, {\mathfrak {a}}}(Z)\) for each \(1 \le t \le s\), whereupon, a straightforward application of Lemma 3.1 delivers the estimate
consequently finishing our proof of Lemma 3.3. \(\square \)
It is worth noting that in the hypotheses of Proposition 3.2 and Lemma 3.3, there are no lower bounds for \(\deg \varphi _i\), whenceforth, we may even choose the polynomials \(\varphi _{1}, \dots , \varphi _{2s}\) to be constant functions. In particular, if \(\varphi _1(x) = \dots = \varphi _{2s}(x) = 1\) for every \(x \in {\mathbb {R}}\), then we see that \(J_{s, {\mathfrak {a}}, \varvec{\varphi }}(A) = M_{s, {\mathfrak {a}}}(A)\) for every finite set \(A \subseteq {\mathbb {Q}}\). This implies that Proposition 3.2 and Lemma 3.3 hold true when the weighted mixed energies \(J_{s, {\mathfrak {a}}, \varvec{\varphi }}(A)\) are replaced by weighted multiplicative energies \(M_{s, {\mathfrak {a}}}(A)\).
4 Chang’s lemma for additive equations and query complexity
Let \(\varphi (x)\) be a polynomial in \({\mathbb {Z}}[x]\) of degree d, for some \(d \in {\mathbb {N}}\). Given finite sets \(A \subseteq {\mathbb {N}}\) and \(V \subseteq {\mathbb {Z}}\) and some prime p, we write
where for any \(n \in {\mathbb {N}}\), the quantity \(\nu _{p}(n)\) denotes the largest exponent \(m \in {\mathbb {Z}}\) such that \(p^m\) divides n. Moreover, we set \(\nu _p(0) = \infty \). When \(V = \{l\}\) for some \(l \in {\mathbb {Z}}\), we write \(A_{p,l} = A_{p,V}\), and we define \(\nu _p(A) = \{ \nu _{p}(a) \ | \ a \in A\}.\) Furthermore, we will frequently use the following straightforward application of Hölder’s inequality, that is, given natural numbers r, s and bounded functions \(f_1, \dots , f_r: [0,1] \rightarrow {\mathbb {C}}\), we have
Our main object of study in this section would be \(E_{s, {\mathfrak {a}}, \varphi }(A)\). We note that these weighted additive energies can be represented as moments of various types of exponential sums, and so, we define the function
when A is a non-empty set and we set \(f_{{\mathfrak {a}}, \varphi }(A; \alpha ) = 0\) if A is an empty set. By orthogonality, we see that
Moreover, note that
which then combines with (4.1) to give us
The following lemma essentially allows us to upgrade the factor \(|\nu _{p}(A)|^{2s-1}\) in the above inequality to a factor of \(O_{s,d}(|\nu _{p}(A)|^{s-1})\).
Lemma 4.1
Let p be a prime number, let \(A \subseteq {\mathbb {N}}\) be a finite set and let \({\mathfrak {a}}: {\mathbb {N}} \rightarrow [0, \infty )\) be a function supported on A. Then
Proof
For ease of notation, we will write \(f(B; \alpha ) =f_{{\mathfrak {a}}, \varphi }(B; \alpha )\) for every \(B \subseteq {\mathbb {Z}}\) and \(\alpha \in {\mathbb {R}}\), thus suppressing the dependence on \({\mathfrak {a}}\) and \(\varphi \). We begin by restricting our analysis to the case when \(\varphi (0) = 0\), since the additive equation \(x_1 + \cdots + x_s = x_{s+1} + \cdots + x_{2s}\) is translation invariant. Thus, let \(\varphi (x) = \sum _{i \in I} \beta _i x^i\) for some non-empty set \(I \subseteq \{1,\dots , d\}\) and for some sequence \(\{\beta _i\}_{i \in I}\) of non-zero integers. Moreover, we define
whereupon, we have \(|X| \le |I|^2 \le d^2\). Denoting \(r = |X|\), we write the elements of X in increasing order as \(x_1< \dots < x_r\), and we decompose the set \(\nu _p(A)\) as \(\nu _p(A) = U_0 \cup \dots U_{r+1}\), where \(U_i = (x_i, x_{i+1}) \cap \nu _p(A)\) for each \(1 \le i \le r-1\) and \(U_0 = (-\infty , x_1) \cap \nu _p(A)\) and \(U_{r} = (x_{r}, \infty ) \cap \nu _p(A)\) and \(U_{r+1} = X \cap \nu _p(A)\). We note that the sets \(U_0, \dots , U_{r+1}\) are pairwise disjoint and that \(|U_{r+1}| \le d^2\).
We begin by applying (4.1) in order to deduce that
Note that if the set \(U_{r+1}\) maximises the right hand side above, then we may apply (4.1) again to deduce the desired claim. Thus, we may assume that \(U_i\) maximises the right hand side above for some \(0 \le i \le r\). We now claim that for any such i there exists \(j = j(i) \in I\) such that for every \(a \in A_{p, U_i}\), we have \(\nu _{p}( \varphi (a)) = \nu _p( \beta _j a^j)\). This arises from combining the fact that
and that m lies in the set \(U_i\), which itself is contained in precisely one of the intervals \((-\infty , x_1), (x_1, x_2), \dots , (x_r, \infty )\).
Now suppose that \(a_1, \dots , a_{2s} \in A_{p, U_i}\) satisfy
Our next claim is that there exist distinct \(k_1, k_2 \in \{1, \dots , 2\,s\}\) such that \(a_{k_1}, a_{k_2} \in A_{p,n}\) for some \(n \in U_i\). If this was not so, then writing \(k_0\) to be the distinct k which minimises \(\nu _{p}(a_k)\), we have that
which delivers a contradiction. Combining this claim with the orthogonality relation (4.2) and applying triangle inequality, we get that
Employing Hölder’s inequality now enables us to bound the right hand side by
and subsequently, we have that
Substituting this into (4.3) finishes our proof. \(\square \)
In order to present the next lemma, we record some further notation, and thus, given a finite set \(A \subseteq {\mathbb {Z}}\), we define the query complexity q(A) to be the minimal \(t \in {\mathbb {N}}\) such that there are functions \(f_1, \dots , f_{t-1}: {\mathbb {Z}} \rightarrow {\mathbb {P}}\) and a fixed prime number \(p_1\) satisfying the fact that the vectors \(\{(\nu _{p_1}(a), \dots , \nu _{p_t}(a))\}_{a \in A}\) are pairwise distinct, where the prime numbers \(p_2, \dots , p_t\) are defined recursively by setting \(p_i = f_{i-1}(\nu _{p_{i-1}}(a))\), for each \(2 \le i \le t\).
Lemma 4.2
Let s, t be natural numbers, let \(A \subseteq {\mathbb {N}}\) be a finite set such that \(q(A) = t\) and let \({\mathfrak {a}}: {\mathbb {N}} \rightarrow [0, \infty )\) be a function supported on A. Then we have that
Proof
For the sake of exposition, we will write \(E(X) = E_{s, {\mathfrak {a}}, \varphi }(X)\) for any subset X of A. Our aim is to prove our lemma by induction on t, and so, we first consider the case when \(t=1\). In this case, there must exist some prime p such that \(\nu _{p}(a)\) is distinct for each \(a \in A\). Applying Lemma 4.1 along with the orthogonality relation (4.2), we infer that
Moreover, since for each \(n \in \nu _p(A)\), the set \(A_{p,n}\) may be written as \(A_{p,n} = \{ a_n\}\) for some unique \(a_n \in A\), we see that \(E(A_{p,n}) = {\mathfrak {a}}(a_n)^{2s}\). Substituting this in the above expression gives the desired bound when \(t=1\).
We now assume that \(t >1\), whence, there exist functions \(f_1, \dots , f_{t-1}: {\mathbb {Z}} \rightarrow {\mathbb {P}}\) and a fixed prime number \(p_1\) satisfying the fact that the vectors \(\{(\nu _{p_1}(a), \dots , \nu _{p_t}(a))\}_{a \in A}\) are pairwise distinct, where the prime numbers \(p_2, \dots , p_t\) are defined recursively by setting \(p_i = f_{i-1}(\nu _{p_{i-1}}(a))\), for each \(2 \le i \le t\). As before, upon applying an amalgamation of Lemma 4.1 and (4.2), we see that
Note that for each \(n \in \nu _{p_1}(A)\), the set \(A_{p_1, n}\) satisfies \(q(A_{p_1, n}) \le t-1\), and so, we may apply the inductive hypothesis for each set \(A_{p_1,n}\) to get the inequality
This finishes the inductive step, and subsequently, the proof of Lemma 4.2 as well. \(\square \)
5 Chang’s lemma for multiplicative equations and an averaging argument
Our main aim of this section is to prove a multiplicative analogue of Lemma 4.2, and this forms the content of the following result.
Lemma 5.1
Let s, d be natural numbers, let \(\varphi \in {\mathbb {Z}}[x]\) be a polynomial of degree d, let A be a finite subset of \({\mathbb {Z}} {\setminus } ( \{0\} \cup {\mathcal {Z}}_{\varphi })\) and let \({\mathfrak {a}}: {\mathbb {R}} \rightarrow [0, \infty )\) be a function. Then
We begin this endeavour by proving an analogue of Lemma 4.2 for \(J_{s, {\mathfrak {a}}, \varphi }(A)\).
Lemma 5.2
Let p be a prime number, let \(\varphi \) be a polynomial of degree d such that \(\varphi (0) \ne 0\), let \({\mathcal {I}} \subseteq {\mathbb {R}}\) be some open interval such that \(xy >0\) and \(\varphi (x) \varphi (y) >0\) for every \(x,y \in {\mathcal {I}}\), let \(A \subseteq {\mathcal {I}}\) be a finite set and let \({\mathfrak {a}}: {\mathbb {N}} \rightarrow [0, \infty )\) be a function. Then we have
Proof
For ease of exposition, we write \(J(A) = J_{s, {\mathfrak {a}}, \varphi }(A)\), thus suppressing the dependence on \(s, {\mathfrak {a}}, \varphi \). Moreover, we import various notation from the proof of Lemma 4.1, and so, suppose that \(\varphi (x) = \sum _{i \in I} \beta _i x^{i}\) for some \(\{0,d\} \subseteq I \subseteq \{0,1,\dots , d\}\) and for some sequence \(\{ \beta _i\}_{i \in I}\) of non-zero integers. We define the elements \(x_1< \dots < x_r\) of the set X, and the sets \(U_0, \dots , U_{r+1}\) precisely as in the proof of Lemma 4.1. The only difference is that in this case, we will have that \(|U_{r+1}| \le r = |X| \le |I|^2 = (d+1)^2\).
Our first claim is that for any fixed \(0 \le i \le r\), whenever \(a_1, \dots , a_{2s} \in A_{p, U_i}\) satisfy
then there exist distinct \(k_1, k_2 \in \{1, \dots , 2s\}\) such that \(\nu _p(a_{k_1}) = \nu _p(a_{k_2})\). We prove this by contradiction, and so, suppose \(a_1, \dots , a_{2s} \in A_{p, U_i}\) satisfy the above system of equations and let k and \(k'\) be the unique elements in \(\{1, \dots , 2s\}\) for which \(\nu _p(a_k)\) is minimal and \(\nu _p(a_{k'})\) is maximal. Moreover, since \(a_1, \dots , a_{2\,s} \in A_{p,U_i}\), there exist distinct \(j_1, j_2 \in I\) such that
for every \(j \in I {\setminus } \{j_1, j_2\}\) and for every \(1 \le l \le 2\,s\). This implies that
where
In the case when \(j_2 > j_1\), utilising the fact that \(a_1 \dots a_s = a_{s+1} \dots a_{2\,s}\), we get
which implies that \(\nu _p(0) < \infty \), thus delivering the desired contradiction. Similarly if \(j_1 > j_2\), then we get that
which subsequently implies that \(\nu _p(0) < \infty \) as well, providing the required contradiction.
With this claim in hand, we now use (3.5) to deduce that
If the maximum is attained by the term corresponding to the set \(A_{p, U_{r+1}}\), we may then combine a second application of (3.5) along with the facts that \(A_{p, U_{r+1}} = \cup _{n \in U_{r+1}}A_{p,n}\) and \(|U_{r+1}| \le r \le (d+1)^2\) to deliver the desired bound. Thus, it suffices to consider the case when the maximum in the preceding expression is attained for some fixed \(i \in \{0,1,\dots , r\}\). In this case, we may use our aforementioned claim to note that
where the maximum is taken over all choices of sets \(B_1, \dots , B_{2\,s}\) such that precisely two of these sets are \(A_{p,n}\) and the rest are \(A_{p, U_i}\). We may now apply (3.4) to deduce that
Combining this with the preceding discussion delivers the bound
which, in turn, combines with the fact that \(r \le (d+1)^2\) give us the desired result. \(\square \)
As in Sect. 4, the above lemma may be iterated to furnish the following result.
Lemma 5.3
Let \(\varphi \) be a polynomial of degree d such that \(\varphi (0) \ne 0\), let \({\mathcal {I}} \subseteq {\mathbb {R}}\) be some open interval such that \(xy >0\) and \(\varphi (x) \varphi (y) >0\) for every \(x,y \in {\mathcal {I}}\), let \(A \subseteq {\mathcal {I}}\) be a finite set such that \(q(A) = t\) and let \({\mathfrak {a}}: {\mathbb {N}} \rightarrow [0, \infty )\) be a function. Then we have
We will now combine this with an averaging argument to deliver bounds for \(M_{s, {\mathfrak {a}}, \varphi }(A)\), and in this endeavour, we first prove a more general lemma that allows us to bound the number of solutions to a system of equations by inserting further auxiliary equations. Thus, given natural numbers \(d_1, d_2\) and functions \(f: {\mathbb {R}} \rightarrow {\mathbb {R}}^{d_1}\) and \(g: {\mathbb {R}} \rightarrow {\mathbb {R}}^{d_2}\) and \({\mathfrak {a}}: {\mathbb {R}} \rightarrow [0, \infty )\), we let
In particular, this counts the number of solutions to the system
where each solution \((x_1, \dots , x_{2s}) \in A^{2s}\) is being counted with weights \({\mathfrak {a}}(x_1) \dots {\mathfrak {a}}(x_{2\,s})\). Similarly, we define
Lemma 5.4
Let \(s, d_1, d_2\) be natural numbers and let \(f: {\mathbb {R}} \rightarrow {\mathbb {R}}^{d_1}\) and \(g: {\mathbb {R}} \rightarrow {\mathbb {R}}^{d_2}\) and \({\mathfrak {a}}: {\mathbb {R}} \rightarrow [0, \infty )\) be functions. Then for any finite set A of real numbers, we have that
We will now show that Lemmata 5.4 and 5.3 imply Lemma 5.1.
Proof of Lemma 5.1
We first partition the set A into finite sets \(A_1, \dots , A_r\), for some \(r \ll _{d} 1\), such that for each \(1 \le j \le r\), the set \(A_j\) lies in some interval \(I_j\) for which we have \(x y >0\) and \(\varphi (x) \varphi (y) >0\) for every \(x,y \in I_j\). Noting the remark following Lemma 3.3, we may apply Lemma 3.3 for the multiplicative energies \(M_{s, {\mathfrak {a}}, \varphi }(A_1 \cup \dots \cup A_r)\), whereupon, we see that it suffices to bound \(M_{s, {\mathfrak {a}}, \varphi }(A_i)\) for each \(1 \le i \le r\) individually. Moreover, for each such \(A_i\), we can let \(f(x) = \log x\) and \(g(x) = \log \varphi (x)\) and apply Lemma 5.4 to deduce that
We obtain the required bound by substituting the estimate presented in the conclusion of Lemma 5.3. \(\square \)
We end this section by recording the proof of Lemma 5.4.
Proof of Lemma 5.4
Let \(Y = \{ (f(a), g(a)) \ | \ a \in A\}\) and let G be the additive abelian group generated by elements of Y. Note that
We now introduce some further notation, and so, given functions \(h_1, h_2: G \rightarrow {\mathbb {R}}\), we define the function \(h_1 * h_2: G \rightarrow {\mathbb {R}}\) by setting
for every \(\varvec{x} \in G\). Next, for every \(s \ge 2\), we define the function \(h_1*_{s} h_1= h_1 * (h_1 *_{s-1} h_1)\) as well as \(h_1*_{1} h_1 = h_1* h_1\). Moreover, for any finite set X, we define the function \({\mathfrak {b}}: G \rightarrow [0, \infty )\) by letting \({\mathfrak {b}}(x,y) = {\mathfrak {a}}(a)\) whenever \((x,y) = (f(a), g(a))\) for some \(a \in A\), and we set \({\mathfrak {b}}(x,y) = 0\) for every other choice of \((x,y) \in G\). Finally, we let \({\mathfrak {b}}': G \rightarrow [0, \infty )\) be a function defined as \({\mathfrak {b}}'(x,y) = {\mathfrak {b}}(-x,-y)\) for every \((x,y) \in G\).
The above notation allows us to rewrite (5.1) as
Applying Young’s convolution inequality, we see that
Double counting then implies that
which subsequently combines with the preceding discussion to give us the bound
thus concluding our proof of Lemma 5.4. \(\square \)
6 Inverse results from additive combinatorics
We utilise this section to present the various inverse results from additive combinatorics that we shall employ in our proof of Theorem 1.2, and we begin this endeavour by recording some definitions. For any subsets A, B of some abelian group G, define
Moreover, throughout this part of the paper, we will denote \({\mathcal {C}}\) to be some large, absolute, computable, positive constant, which may change from line to line.
First, we will record a classical result in additive combinatorics known as the Plünnecke–Ruzsa inequality [27, Corollary 6.29], see also [17] for a shorter and simpler proof.
Lemma 6.1
Let A be a finite subset of some additive abelian group G. If \(|A+A| \le K|A|\), then for all non-negative integers m, n, we have
We now present one of the main inverse results that we will use to prove Theorem 1.2.
Theorem 6.2
Let \(A \subseteq {\mathbb {R}}\) be a finite set, let \(s\ge 16\) be an integer and let \(E_{s}(A) = |A|^{2s-1}/K\), for some \(K \ge 1\). Then there exists \(2 \le s' \le s\) and a finite, non-empty set \(U' \subseteq s'A\) satisfying
such that for every \(m,n \in {\mathbb {N}} \cup \{0\}\), we have
We will prove this by utilising some of the ideas from [14], and in particular, we will use the following result that may be deduced from the proof of Proposition 2.3 in §8 of [14].
Proposition 6.3
Let \(\nu , \delta \) be positive real numbers such that \(\nu \ge 1\) and let \(s \ge 4\) be some even number. Moreover, suppose that \(A \subseteq {\mathbb {R}}\) is a finite, non-empty set such that \(E_{s}(A) \ge |A|^{2s - \nu }\). Then we either have \(E_{s/2}(A) > |A|^{s - \nu + \delta },\) or there exists some non-empty subset \(U' \subseteq (s/2)A\) satisfying
such that for every \(m, n \in {\mathbb {N}} \cup \{0\}\), we have
We are now ready to prove Theorem 6.2.
Proof of Theorem 6.2
Letting \(l \ge 4\) be the integer satisfying \(2^l \le s < 2^{l+1}\), we may use Lemma 3.1 to deduce that
where we write \(|A|^{\nu -1} = K\). In particular, this means that
whereupon, there exists some \(2 \le i \le l\) such that
Defining \(\nu '\) to be the real number such that \(E_{2^i}(A) =|A|^{2^{i+1} - \nu '}\), we see that \(\nu \ge \nu ' \ge 1\) since
Furthermore, by the preceding discussion, we have that
We may now apply Proposition 6.3 with \(s = 2^i\) and \(\delta = (\nu -1)/(l-1)\) to deduce the existence of some finite, non-empty set \(U' \subseteq 2^{i-1}A\) satisfying
such that for every \(m, n \in {\mathbb {N}} \cup \{0\}\), we have
We obtain the desired conclusion by noting that \((\log s)/2 \le l \le \log s\) and substituting \(|A|^{\nu - 1} = K\). \(\square \)
Theorem 6.2 can be seen as an efficient many-fold version of a Balog–Szemerédi–Gowers type theorem, see [23] for more details about the latter. We further note that a multiplicative analogue of Theorem 6.2 for finite sets \(A \subseteq {\mathbb {N}}\) follows in a straightforward manner from Theorem 6.2 by considering the logarithmic map from \({\mathbb {N}}\) to \([0, \infty )\).
Our next goal in this section is to prove the following lemma which arises from adapting some of the methods from [16], see also [8].
Lemma 6.4
Let A, X be finite, non-empty subsets of \({\mathbb {N}}\) such that \(|A \cdot X\cdot X| \le K|X|\), for some \(K \ge 1\). Then there exists a subset \(B \subseteq A\) such that
In our proof of the above result, we will closely follow ideas and definitions as recorded in [8, Chapter 8], and so, given \(r \in {\mathbb {N}}\), we define the map \(\pi _{i,r}: {\mathbb {Z}}^r \rightarrow {\mathbb {Z}}\) as \(\pi _{i,r}(x_1, \dots , x_r) = x_i\) for every \((x_1, \dots , x_r) \in {\mathbb {Z}}^r\). We denote a finite set \(X \subseteq {\mathbb {Z}}\) to be a quasicube if \(|X|=2\). Moreover, when \(r \ge 2\) and \(X \subseteq {\mathbb {Z}}^r\) is some finite set, then we write X to be a quasicube if \(\pi _{r,r}(X) = \{y_1, y_2\}\) for some distinct \(y_1, y_2 \in {\mathbb {Z}}\) as well as if the sets
and
are also quasicubes. Moreover, a subset of a quasicube is called a binary set.
We define a set \(V \subseteq {\mathbb {Z}}^r\) to be an axis aligned subspace if \(V = X_1 \times \dots \times X_r\), where for every \(1 \le i \le r\), we either have \(X_i = \{x_i\}\) for some \(x_i \in {\mathbb {Z}}\) or \(X_i = {\mathbb {Z}}\). Moreover, if \(\pi _{i,r}(V)\) is not a singleton for some \(1 \le i \le r\), then we call \(\pi _{i,r}\) to be a coordinate map on V. Finally, given a finite subset X of some axis aligned subspace V, we will now define its skew-dimension \(d_*(X)\). If \(|X| = 1\), then we denote \(\dim _*(X) = 0\). Otherwise, let \(1 \le i \le r\) be the largest number such that \(\pi _{i,r}\) is a coordinate map on V and \(|\pi _{i,r}(A)| >1\). In this case, we define
With this notation in hand, we are now ready to prove Lemma 6.4.
Proof of Lemma 6.4
Since \(A, X \subseteq {\mathbb {N}}\) are finite sets, we note that the set
is finite, whence, writing \({\mathcal {P}} = \{ p_1, \dots , p_r\}\) for some \(r \in {\mathbb {N}}\), we define the map \(\psi : {\mathbb {N}} \rightarrow {\mathbb {Z}}^d\) such that \(\psi (y) = (\nu _{p_1}(y), \dots , \nu _{p_r}(y))\). For the purposes of this proof, let \(\pi _i: {\mathbb {Z}}^r \rightarrow {\mathbb {Z}}\) be the projection map defined as \(\pi _i(x_1, \dots , x_r) = x_i\) for every \(1 \le i \le r\). Our hypothesis now implies that \(|\psi (A) + \psi (X) + \psi (X)| \le K |\psi (X)|\), and so, writing V to be the largest binary set contained in \(\psi (A)\), we use [13, Theorem 2.7] to deduce that
But now, we may employ [8, Proposition 8.4.2] to infer that there exists \(A' \subseteq A\) such that
Noting the definition of skew-dimension and query complexity, it is relatively straightforward to show that \(q(A') \le d_{*}(\psi (A')) +1\). Combining this with the preceding discussion dispenses the desired conclusion. \(\square \)
A key ingredient in our proof of Theorem 1.2 will be iterative applications of Theorem 6.2. In particular, assuming our set A to have a large multiplicative energy, we will apply Theorem 6.2 to extract a large subset of A that exhibits various other types of multiplicative structure. Moreover, we keep repeating this argument until the remaining set has a small multiplicative energy. Here, it will be important to be able to control the number of steps that such an iterative process takes, a task for which we will employ [14, Lemma 4.2]. We present this below.
Lemma 6.5
Let \(0< c < 1\) and \(C>0\) be constants. Let \(A_0 = A\), and for each \(i \ge 1\), define \(A_i = A_{i-1} {\setminus } U_i\) where \(U_i\) is some set satisfying \(|U_i| \ge C|A_{i-1}|^{1-c}\). Then, for some \(r \le 2 (\log |A| + 2) + C^{-1} \frac{|A|^{c}}{2^c - 1}\), we must have \(|A_r| \le 1\).
7 Proof of Theorem 1.2
We dedicate this section to proving Theorem 1.2. Let \({\mathcal {D}}\) be some large constant that we will fix later, and let s be some natural number sufficiently large in terms of \({\mathcal {D}}\). We define
We first begin with assuming that our set A is a subset of \({\mathbb {N}} {\setminus } {\mathcal {Z}}_{\varvec{\varphi }}\). Furthermore, we may assume that \(M_{s}(A) > |A|^{2s - k}\), since otherwise, we are done.
As previously mentioned, we now perform an iteration, where at each step, we extract a large subset of our set A which satisfies suitable arithmetic properties. We first set \(A_0 = A\). Next, for every \(i \in {\mathbb {N}}\), we will begin the ith step of our iteration with some subset \(A_{i-1} \subseteq A_0\). If \(M_{s}(A_{i-1}) \le |A_{i-1}|^{2s - k}\), we stop our iteration, else, we apply a multiplicative version of Theorem 6.2 to obtain a finite, non-empty subset \(U_i \subseteq {\mathbb {N}}\) satisfying
such that
holds true for every \(m,n \in {\mathbb {N}} \cup \{0\}\). In particular, writing \(A_{i-1}' = A_{i-1} \cap x^{-1} \cdot U_i\), for some \(x \in U_i \cdot A_{i-1}^{-1}\) which maximises \(|A_{i-1} \cap x^{-1} \cdot U_i|\), we get that \(|A_{i-1}'| \gg |A_{i-1}|^{ 1- k {\mathcal {C}}/ \log s}\). Thus we see that
which then combines with Lemma 6.4 to give us a subset \(B_i \subseteq A_{i-1}'\) such that
Moreover, since \(B_i \subseteq x^{-1} \cdot U_i\), we may also deduce that
for every \(m, n \in {\mathbb {N}} \cup \{0\}\). We now set \(A_i = A_{i-1} {\setminus } B_i\) and proceed with the \((i+1)\)th step of our algorithm. By way of Lemma 6.5, we must have \(|A_r| \le 1\) for some
in which case, we would trivially have \(M_{s}(A_r) \le |A_r|^{2\,s - k}\), thus terminating the algorithm. This gives us a partition of the set A as \(A = B_1 \cup \dots \cup B_{r} \cup A_r\), where the sets \(A_r, B_1, \dots , B_r\) are pairwise disjoint. We set \(C = A_r\) and \(B = B_1 \cup \dots \cup B_r\).
We first proceed to prove the bound on the mixed multiplicative energy \(M_{s, \varvec{\varphi }}(B)\), and so, just for this part of the proof, we assume that \(\varphi _j(0) \ne 0\) for \(1 \le j \le 2s\). Writing \(q = 10k\), this allows us to employ Lemma 5.1 to see that
for each \(1 \le i \le r\) and \(1 \le j \le 2\,s\). We may combine this with (7.2) and (7.3) so as to obtain the bound
Next, we use the remark following Lemma 3.3 along with the preceding inequalities to deduce that
It is worth noting that
whenever \({\mathcal {D}}\) is sufficiently large in terms of \({\mathcal {C}}\). Thus, the estimate
holds true whenever \({\mathcal {D}}\) is sufficiently large in terms of \({\mathcal {C}}_d\). Noting the fact that for any \(x \in {\mathbb {R}}\) and any \(\varphi \in {\mathbb {Z}}[x]\) satisfying \(1 \le \deg \varphi \le d\), there are at most \(O_d(1)\) solutions to \(x = \varphi (b)\) with \(b \in B\), we deduce that
This combines with the preceding discussion to give us
where the second and third inequalities follow from the analogues of Proposition 3.2 and Lemma 3.3 mentioned in the remark at the end of Sect. 3 respectively.
We now turn to the case of dealing with the mixed additive energy \(E_{s, \varvec{\varphi }}(B)\). This, in fact, proceeds in a very similar fashion to the proof for the upper bound on \(M_{s, \varvec{\varphi }}(B)\), just with the applications of Lemmata 5.1 and 3.3 replaced by applications of Lemmata 4.2 and 3.1 respectively. In fact, the case of \(E_{s, \varvec{\varphi }}(B)\) is slightly simpler than the situation where we provide estimates for \(M_{s, \varvec{\varphi }}(B)\), since the former does not require utilising the upper bound (7.3) on the many-fold product sets of \(B_1, \dots , B_r\).
Thus, we have proven Theorem 1.2 when A is a finite subset of \({\mathbb {N}}{\setminus } {\mathcal {Z}}_{\varvec{\varphi }}\) and \(\varvec{\varphi } \in ({\mathbb {Z}}[x])^{2s}\). We now reduce the more general case when A is a finite subset of \({\mathbb {Q}}\) and \(\varvec{\varphi } \in ({\mathbb {Q}}[x])^{2s}\) to the aforementioned setting. As in Sect. 3, we see that upon dilating the set A appropriately, we may reduce the more general case to the setting when \(A \subseteq {\mathbb {Z}}\) and \(\varvec{\varphi } \in ({\mathbb {Z}}[x])^{2\,s}\). We now focus on the multiplicative setting of Theorem 1.2 for the latter case, and so, given some finite \(A \subseteq {\mathbb {Z}}\) and some \(\varvec{\varphi } \in ({\mathbb {Z}}[x])^{2\,s}\) such that \(\varphi _{j}(0) \ne 0\) for every \(1 \le j \le 2\,s\), we define \(\varvec{\varphi }'\) to satisfy \(\varphi _j'(x)= \varphi _j(-x)\) for every \(x \in {\mathbb {Z}}\) and \(1 \le j \le 2\,s\). Next, we write \({\mathcal {Z}} = ({\mathcal {Z}}_{\varvec{\varphi }} \cup {\mathcal {Z}}_{\varvec{\varphi }'}){\setminus } \{0\}\) and we denote
By Theorem 1.2, we see that \(A_1 = B_1 \cup C_1\) and \(-A_2 = (-B_2) \cup (-C_2)\) such that for every \(i \in \{1,2\}\), we have \(B_i \cap C_i = \emptyset \) as well as
and
for some \(\eta _s \gg _{d} \log s/ \log \log s\). Noting that \(M_{s, {\varphi }_j'}(-B_2) = M_{s, {\varphi }_j}(B_2)\) for every \(1\le j\le 2\,s\) and \(|A_3|,|A_4| \ll _{d} 1\), we may now write \(B = B_1 \cup B_2 \cup A_4\) and \(C = C_1 \cup C_2 \cup A_3\), whereupon, the remark following Lemma 3.3 implies that
A similar strategy maybe followed to resolve the case when \(M_{s, \varvec{\varphi }}(B)\) is replaced by \(E_{s, \varvec{\varphi }}(B)\). This concludes our proof of Theorem 1.2.
8 Proof of Theorems 1.6 and 2.1
Our aim in this section is to prove Theorem 1.6 and its multiplicative variants. We begin by recording the following greedy covering lemma from [9, Lemma 5].
Lemma 8.1
Let A, B be subsets of \({\mathbb {R}} {\setminus } \{0\}\) such that \(|A| \ge 2\) and \(|A \cdot B| \le C|B|\), for some \(C \ge 1\). Then there exists a set \(S \subseteq A \cdot B^{-1}\) with \(|S| \ll C \log |A|\) such that \(A \subseteq S \cdot B\).
We are now ready to prove Theorem 1.6.
Proof of Theorem 1.6
We begin noting that Theorem 1.6 holds trivially when \(|A|=1\), and so, we may assume that \(|A| \ge 2\). We now apply Lemma 6.1 to infer that \(|A^{(3)}| \le K^3 |A|\), whence, we may apply Lemma 6.4 to deduce the existence of a set \(A' \subseteq A\) with \(|A'| \ge |A|/K^3\) and \(q(A') \le 3 \log K + O(1)\). This implies that \(|A \cdot A'| \le K|A| \le K^{4} |A'|\), whenceforth, we may apply Lemma 8.1 to obtain a set \(S \subseteq {\mathbb {Q}} {\setminus } \{0\}\) such that \(|S| \ll K^{4} \log |A|\) and \(A \subseteq S \cdot A'\). In particular, let \(S = \{s_1, \dots , s_r\}\), where \(r = |S|\), and let \(A_i = A \cap s_i \cdot A'\). Using orthogonality and applying Hölder’s inequality as in (4.1), we may deduce that
Finally, since for each \(1 \le i \le r\), the set \(A_i\) lies in a dilate of the set \(A'\), we must have \(q(A_i) \le q(A') \le 3 \log K + O(1)\). This allows us to apply Lemma 4.2 to deduce that
where \(C= 8 + 12 \log (d^2 + 2) + 6 \log (2s)\). \(\square \)
The proof of Theorem 2.1 follows mutatis mutandis, and we make some brief remarks concerning this below. Adapting the methods of Sect. 5 along with the results of this section, that is, following ideas from Sect. 5 while replacing any usage of Lemma 4.1 with Lemma 5.3 as well as substituting the application of Hölder’s inequality in (8.1) with an application of Lemma 3.3, we may obtain a similar result for \(J_{s, {\mathfrak {a}}, {\varphi }}(A)\), where \(\varphi \in {\mathbb {Q}}[x]\) with \(\varphi (0) \ne 0\). We may further use a combination of Lemmata 5.1 and 6.1 in place of Lemma 5.3 to obtain an estimate for \(M_{s, {\mathfrak {a}}, {\varphi }}(A)\), when \(|A\cdot A| \le K |A|\). This finishes the proof of Theorem 2.1 when \(\varvec{\varphi } = (\varphi , \dots , \varphi )\) for some \(\varphi \in {\mathbb {Q}}[x]\) with \(\varphi (0) \ne 0\). The more general case may then be deduced from this special case by employing Proposition 3.2 and the ideas involved therein.
9 Additive and multiplicative Sidon sets
We dedicate this section to proving Theorem 1.4 and Proposition 1.5. In our proof of Theorem 1.4, we will use the following amalgamation of [11, Lemmata 5.1 and 5.2].
Lemma 9.1
Given a finite set \(A \subseteq {\mathbb {N}}\) and natural numbers \(s \ge 2\) and \(d \ge 1\) and some real number \(c>0\), if \(E_{s}(A) \ll _{d,s} |A|^{2\,s - 2 + 1/s - c}\), then there exists a \(B_{s}^{+}[1]\) set \(X \subseteq A\) with \(|X| \gg _{d,s} |A|^{1/s +c/2\,s}\). Similarly, if we have \(M_{s}(A) \ll _{d,s} |A|^{2\,s - 2 + 1/s - c}\), then there exists a \(B_{s}^{\times }[1]\) set \(Y \subseteq A\) satisfying \(|Y| \gg _{d,s} |A|^{1/s + c/2s}\)
With this result in hand, we now present the proof of Theorem 1.4.
Proof of Theorem 1.4
We begin by applying Theorem 1.2 to obtain a decomposition \(A = B \cup C\), with B, C being disjoint and satisfying the conclusion of Theorem 1.2. We divide our proof into two cases, the first of these being if \(|C| \ge |A|/2\). Writing \(C_1 = C \cup (0, \infty )\) and \(C_2 = C \cup (-\infty , 0)\), we see that either \(|C_1| \ge |C|/3\) or \(|C_2| \ge |C|/3\). If the first inequality holds, then we see that
whence, applying Lemma 9.1 delivers a \(B_{s}^{\times }[1]\) subset \(Y \subseteq C\) such that
On the other hand, if \(|C_2| \ge |C|/3\), then we may again employ Lemma 9.1 to obtain a large \(B_{s}^{\times }[1]\) subset \(Y'\) of \(-C_2\), which, in turn, gives us a large \(B_{s}^{\times }[1]\) subset \(-Y'\) of \(C_2\).
On the other hand, if \(|B| \ge |A|/2\), then we may define \(B_1 = \{ b \in B \ | \ \varphi (b) \in (0, \infty )\}\) and \(B_2 = \{ b \in B \ \ \varphi (b) \in (- \infty ,0)\}\). As before, either \(|B_1| \ge |B|/3\) or \(|B_2| \ge |B|/3\). If the first inequality holds, then we have that
whence Lemma 9.1 yields a \(B_{s}^{+}[1]\) set \(X' \subseteq \varphi (B_1)\) such that \(|X'| \gg _{s,d} |\varphi (B_1)|^{\eta _s/4\,s}\). For every \(x \in X'\), fix some \(b_x \in B\) such that \(\varphi (b_x) = x\) and let \(X = \{ b_x \ | \ x \in X' \}\). Then we have that
and so, we finish the subcase when \(|B_1| \ge |B|/3\). We may proceed similarly in the case when \(|B_2| \ge |B_1|/3\) to finish the proof of additive part of Theorem 1.4. As for the multiplicative case, since \(\varphi (0) \ne 0\), we see that \(M_{s, \varphi }(B) \ll _{s,d} |B|^{2\,s - \eta _s}\). We may now continue as in the additive case to finish our proof of Theorem 1.4. \(\square \)
Thus, we have shown, in the form of Theorem 1.4, that strong low energy decompositions deliver large additive and multiplicative Sidon sets. We will now show that, roughly speaking, such an implication may be reversed as well.
Proposition 9.2
Let \(s \ge 2\), let \(\varvec{\varphi } \in ({\mathbb {Z}}[x])^{2\,s}\) satisfy \(\varphi _1 = \dots = \varphi _{2s} = \varphi \), for some \(\varphi \in {\mathbb {Z}}[x]\) with \(\deg \varphi = d \ge 1\) and \(\varphi (0) \ne 0\), and let \(\delta _s\) be defined as in Theorem 1.4. Then, given any finite set \(A \subseteq {\mathbb {Z}}\), we can find disjoint sets \(B, C \subseteq A\) such that \(A = B \cup C\) and
Proof
Let A be a finite set of integers. We may iteratively apply Theorem 1.4 to the set A to obtain an absolute constant \(D = D_{d,s}>0\) and sets \(A =A_0 \supseteq A_1 \supseteq \dots \supseteq A_r\), with \(|A_{i-1} {\setminus } A_i| = D |A_{i-1}|^{\delta _s/s}\) for every \(1 \le i \le r\), where the set \(A_{i-1} {\setminus } A_i\) is either a \(B_{s}^{+}[1]\) set or a \(B_{s}^{\times }[1]\) set. A straightforward application of Lemma 6.5 implies that \(|A_r| \le 1\) for some \(r \ll _s |A|^{1 - \delta _s/s}\). Thus, we may partition A as
where \(X_i\) is a \(B_{s, \varphi }^{+}[1]\) set for every \(1 \le i \le r_1\), the set \(Y_j\) is a \(B_{s}^{\times }[1]\) set for every \(1 \le j \le r_2\), the sets \(X_1, \dots , X_{r_1}, Y_1, \dots , Y_{r_2}\) are pairwise disjoint and \(r_1 + r_2 \ll _s |A|^{1 - \delta _s/s}\). Writing \(B = \cup _{1 \le i \le r_1} X_i\), we may apply Lemma 3.1 to deduce that
One may similarly write \(C = \cup _{1 \le j \le r_2} Y_j\) and apply Lemma 3.3 suitably to obtain the bound \(M_{s}(C) \ll _{s,d} |A|^{2\,s - \delta _s}\). Finally, applying this circle of ideas mutatis mutandis, with \(M_{s, \varvec{\varphi }}(B)\) replacing \(E_{s, \varvec{\varphi }}(B)\), allows us to conclude the proof of Proposition 9.2. \(\square \)
We end this section by recording the proof of Proposition 1.5.
Proof of Proposition 1.5
Let \(d,s \in {\mathbb {N}}\) satisfy \(s \ge 10d(d+1)\) and write
for some \(n,m \in {\mathbb {N}}\) to be fixed later. We will first show that for appropriate choices of m, n, every \(B \subseteq A\) with \(|B| \ge |A|/2\) and every \(\varphi \in {\mathbb {Z}}[x]\) with \(\deg \varphi = d\) satisfy
where \(\varvec{\varphi } = (\varphi , \dots , \varphi )\). We begin by observing that \(|B^{(s)}| \le |A^{(s)}| \ll _{s} m^{s} n\), which, together with (1.3), gives us
We now turn to analysing \(E_{s, \varvec{\varphi }}(B)\), and so, we write \({S}_j = \{ 2^j \cdot (2i+1) \ | \ 1 \le i \le m \}\) and note that \(A = S_1 \cup \dots \cup S_n\) as well as that \(|A| = mn\). Next, setting
we deduce that
Furthermore, writing \(B_j = B \cap S_j\) for every \(j \in {\mathcal {M}}\) and \(\varphi (x) = a_0 + a_1 x + \cdots + a_d x^d\) for some \(a_0, \dots , a_d \in {\mathbb {Z}}\) with \(a_d \ne 0\), we discern that the set \(s \varphi (B_j)\) is contained in \(s \varphi (S_j)\), which itself is a subset of the set
In particular, we have that
whence, applying (1.3) again gives us
Combining this with (9.2) then allows us to infer that
We may now optimise this with the inequality recorded in (9.1), which gives us that \(m^{s - d(d+1)/2} = n^{2\,s - 2}\), that is, \(n^{3\,s - d(d+1)/2 - 2} = |A|^{s - d(d+1)/2}\). Here, choosing n to be appropriately large in terms of N already gives us \(|A| \gg N\). Moreover, substituting the value of n into the preceding set of inequalities yields the bound
An elementary computation now shows that
whenever \(s \ge 10d(d+1)\), consequently proving the first part of Proposition 1.5.
For our second part, let \(s \ge 2\) be an even integer and let P be the set consisting of the first \(\lceil N^{\frac{s}{2s+2}}\rceil \) prime numbers and let Q be the set consisting of the next \(\lceil N^{\frac{s+2}{2\,s+2}}\rceil \) prime numbers. We further define \(A':= A_{N}' = P \cdot Q,\) whence, we have the estimate \(|A'| \gg N\). In [11, Section 2], applying graph theoretic results from [15], it was shown that the largest \(B_{s}^{\times }[1]\) subset Y of \(A'\) satisfies \(|Y| \ll _{s} |A'|^{\frac{1}{2} + \frac{1}{s+2}}\), thus, we only focus on the additive case here. By way of the prime number theorem, we may deduce that \(A' \subseteq \{1, \dots , M\}\) for some \(M \ll N (\log N)^2\), whence, writing X to be the largest \(B_{s, \varphi }^{+}[1]\) subset of \(A'\), we get that
This dispenses the desired bound, and so, we conclude our proof of Proposition 1.5. \(\square \)
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Acknowledgements
The author is supported by Ben Green’s Simons Investigator Grant, ID 376201. The author is grateful to Ben Green and Oliver Roche-Newton for helpful discussions. The author would like to thank the anonymous referee for a careful reading of the manuscript and for various helpful comments.
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