Unbounded expansion of polynomials and products

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 $1 \leq deg \varphi_i \leq d$ for every $1 \leq i \leq s$, we prove that \[ |A^{(s)}| + |\varphi_1(A) + \dots + \varphi_s(A) | \gg_{s,d} |A|^{\eta_s} , \] for some $\eta_s \gg_{d} \log s / \log \log s$. Moreover if $\varphi_i(0) \neq 0$ for every $1 \leq i \leq s$, then \[ |A^{(s)}| + |\varphi_1(A) \dots \varphi_s(A) | \gg_{s,d} |A|^{\eta_s}. \] These generalise and strengthen previous results of Bourgain--Chang, P\'{a}lv\"{o}lgyi--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.


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 ⊆ R, we define the s-fold sumset sA and the s-fold product set A (s) of A to be sA = {a 1 + • • • + a s | a 1 , . . ., a s ∈ A} and A (s) = {a 1 . . .a s | a 1 , . . ., a s ∈ A} respectively.These can be seen to be standard measures of arithmetic structure, since whenever A is an arithmetic progression, then |A| ≤ |sA| ≪ s |A|, while whenever A is a geometric progression, we have that |A| ≤ |A (s) | ≪ 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 ∈ N and ε > 0 and finite set A ⊆ 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 |sA| + |A (s) | ≫ s |A| (log s) 1/4 (1.1) 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 → ∞ as s → ∞.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].
When ϕ 1 (x) = • • • = ϕ 2s (x) = x for every x ∈ Z, we write E s (A) = E s,ϕ (A) and M s (A) = M s,ϕ (A).A standard application of the Cauchy-Schwarz inequality then gives us that for every s ∈ N and every finite A ⊆ Z. Noting this along with Conjecture 1.1, it is natural to expect that for every finite A ⊆ Z, one may write A = B ∪ C, with B, C disjoint, such that E s (B) ≪ s |A| 2s−1−cs and M s (C) ≪ s |A| 2s−1−cs , (1.4) 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 → ∞ as s → ∞.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 → ∞ as s → ∞.In our recent work [14], this speculation was confirmed quantitatively with [14,Corollary 1.3] delivering the bound c s ≫ (log log s) 1/2−o (1) .The latter was further quantitatively improved by Shkredov [25] to c s ≫ (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 ≪ 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.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 ⊆ R, we see that since max{|B|, |C|} ≥ |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 ∈ N and any ϕ 1 , . . ., ϕ s ∈ Q[x] such that 1 ≤ deg ϕ i ≤ d for each 1 ≤ i ≤ s and for any finite set A ⊆ Q, we have that for some η s ≫ d log s/ log log s.Moreover if ϕ i (0) = 0 for every 1 ≤ i ≤ s, then Note that upon setting ϕ 1 , . . ., ϕ 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 ϕ 1 , . . ., ϕ 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, for some fixed c, d ∈ Z \ {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 ϕ ∈ Z[x], we denote a finite set X ⊆ Z to be a B + s,ϕ [1] set if for every n ∈ Z, there is at most one distinct solution to the equation with x 1 , . . ., x s ∈ 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,ϕ [1] set if for every n ∈ Z, there is at most one distinct solution to the equation n = ϕ(x 1 ) . . .ϕ(x s ), with x 1 , . . ., x s ∈ X.When ϕ equals the identity map, then a B + s,ϕ [1] set is known as an additive Sidon set, which we will denote as a B + s [1] set.We define a B × s [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, . . ., N}, this has recently been generalised to the setting of finding B + s [1] and B × s [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 [12,18].For instance, it is known that any finite set A ⊆ Z contains a B + s [1] set of size at least c ′ s |A| 1/s for some c ′ s > 0 (see [11], [22]), and this is sharp, up to multiplicative constants, by setting A = {1, . . ., N}.On the other hand, {1, . . ., N} contains large B × s [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 [12] 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 ∈ N, there exists a parameter δ s ≫ d log s/ log log s such that the following holds true.Given any finite set A ⊆ Z and any ϕ The first such result was proven in [12], 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 [1] set of size at least |A| δs/s , with δ s ≫ (log log s) 1/2−o (1) .This was subsequently improved by Shkredov [25] who gave the bound δ s ≫ (log s) 1/2−o (1) .Theorem 1.4 not only quantitatively improves this to δ s ≫ (log s) 1−o (1) , but it also generalises this to estimates on B + s,ϕ [1] and B × s,ϕ [1] sets, for suitable choices of ϕ ∈ 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 (1) .
As before, this recovers the best known result towards Conjecture 1.1 for large values of s by simply setting ϕ 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 η s < s and δ 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 ⊆ A with |B| ≥ |A|/2, we have that thus implying that η s ≤ (2s + 1)/3.Similarly, in the case when ϕ is set to be the identity map, it was noted by Roche-Newton that these sets also gave the bound δ 2 ≤ 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 δ 2 ≤ 4/3.The latter constructions were then generalised in [12,Proposition 1.5] for every s ∈ N, thus giving the estimate δ s /s ≤ 1/2 + 1/(2s + 2) when s is even and δ s /s ≤ 1/2 + 1/(2s) when s is odd.
We now return to the even more general case where s, d are some natural numbers and ϕ ∈ Z[x] is some arbitrary polynomial with deg ϕ = d.In this setting, we are able to utilise ideas from [1] and [12] to record the following upper bounds.
Similarly, when s is even, there exists a finite set This, along with Theorems 1.2 and 1.4, implies that whenever s ≥ 10d(d + 1), then (log s) 1−o(1) ≪ d η s ≤ 2s/3 + (d 2 + d + 2)/6 and (log s) 1−o(1) ≪ d δ s ≤ s/2 + 1/2 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 [12,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 ⊆ Z satisfies |A (2) | ≤ K|A| with K = |A| ηs/10s , then the sumset ϕ 1 (A) + • • • + ϕ 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(θ) = e 2πiθ for every θ ∈ R.
Theorem 1.6.Let K ≥ 1 be a real number and let d, s be natural numbers.Moreover, let A ⊆ Z be a finite set such that |A When ϕ(x) = x for every x ∈ Z, such types of results have been previously referred to as the weak Erdős-Szemerédi Conjecture, which suggests that whenever |A (2) | ≤ K|A|, then we have |2A| ≫ ε |A| 2−ε /K O (1) for every ε > 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,a,ϕ (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 ≤ i ≤ s.
On the other hand, given ε > 0, in the case when |A (2) | ≪ s,d |A| cε |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| ε .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, . . ., A) = {F (a 1 , . . ., a s ) | a 1 , . . ., a s ∈ A}, where F is an arbitrary polynomial in s variables.Estimates for |F (A, . . ., 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 ≤ 1 such that either |F (A, . . ., A)| ≫ |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 , . . ., x s ) = ϕ 1 (x 1 ) + • • • + ϕ s (x s ) or F (x 1 , . . ., x s ) = ϕ 1 (x 1 ) . . .ϕ s (x s ), where ϕ 1 , . . ., ϕ s are univariate polynomials.On the other hand, our results show that over Q, if F is of the latter type with suitably chosen ϕ 1 , . . ., ϕ s , then either |F (A, . . ., 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, §2 is dedicated to presenting some further applications of our method.In §3, we record some properties of the mixed energies E s,ϕ (A) and M s,ϕ (A) that we will use throughout our paper.The first main step towards proving Theorem 1.2 is initiated in §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 ϕ ∈ Z[x] with deg ϕ = d, and finite sets A 0 , A 1 , . . ., A r of natural numbers such that for every 1 ≤ i ≤ r, we have , where a : N → [0, ∞) 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,a,ϕ (A) in terms of the l 2 -norm of a and the so-called query-complexity q(A) of A, see Lemma 4.2 for more details.
We now proceed to §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 J s,a,ϕ (A) = a 1 ,...,a 2s ∈A a(a 1 ) . . .a(a 2s )1 a 1 ...as=a s+1 ...a 2s 1 ϕ(a 1 )...ϕ(as)=ϕ(a s+1 )...ϕ(a 2s ) , and prove decoupling type inequalities for such quantities.Iterating these estimates, as in §4, leads to suitable bounds for J s,a,ϕ (A).Next, we prove a multiplicative variant of an averaging argument from analytic number theory, which allows us to discern bounds on M s,a,ϕ (A) from estimates for J s,a,ϕ (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 §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) ≥ |A| 2s−k , for suitable values of s, k, then A has a large intersection with a set U ′ satisfying |U ′ | ≪ |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 ⊆ N satisfying the inequality |A • X • X| ≤ K|X|, there must exist a large subset B of A with q(B) ≤ 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 ⊆ Z with a large multiplicative energy has a large subset B with a small query complexity.In §7 we apply this idea iteratively, along with the results recorded in § §3-6, to yield the proof of Theorem 1.2.We utilise §8 to prove Theorems 1.6 and 2.1, and finally, in §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 ≫ z Y , or equivalently Y ≪ z X, to mean X ≥ C z |Y | where C is some positive constant depending on the parameter z.We use e(θ) to denote e 2πiθ for every θ ∈ R.Moreover, for every natural number k ≥ 2 and for every non-empty, finite set Z, we use |Z| to denote the cardinality of Z, we write All our logarithms will be with respect to base 2.
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.

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,ϕ (A) and J s,a,ϕ (A), and so, for every ϕ ∈ Q[x] 2s , every a : R → [0, ∞) and every finite set A ⊆ R, we define M s,a,ϕ (A) = a 1 ,...,a 2s ∈A a(a 1 ) . . .a(a 2s )1 ϕ 1 (a 1 )...ϕs(as)=ϕ s+1 (a s+1 )...ϕ 2s (a 2s ) and 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,a,ϕ (A) and M s,a,ϕ (A), whenever A • A is small.
Here, the condition A ⊆ Q\(Z ϕ ∪{0}) seems to be necessary.In order to see this, we may choose ϕ ∈ Q[x] to be some linear polynomial, ϕ = (ϕ, . . ., ϕ) and A = {2, 4, . . ., 2 N } ∪ {x} for some x ∈ Z ϕ .In this case, since ϕ(a 1 ) . . .ϕ(a s−1 )ϕ(x) = ϕ(a s+1 ) . . .ϕ(a 2s−1 )ϕ(x) = 0 for any a 1 , . . ., a s−1 , a s+1 , . . ., a 2s−1 ∈ A, we see that 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 ∈ C \ {0} and for any finite subset 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 ∈ Q \ {0} and for any ε > 0 and for some constant C ε > 0. Using Theorem 2.1, we can prove a non-linear analogue of the above result.
Proof.We begin by partitioning R as , where r ≪ d 1 and I 1 , . . ., I r are open intervals such that the derivative ϕ ′ is non-zero on each such interval and such that the set Next, given any two finite sets X, Y , we denote r(X, Y ) to be the number of solutions to the equation x = ϕ(y), with x ∈ X, y ∈ Y .With this notation in hand, we note that for any 1 since fixing either of x, y in the equation x = ϕ(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 ≤ i, j ≤ r.Fixing some 1 ≤ i, j ≤ r, we define S = {a ∈ A j : ϕ(a) ∈ A i }, whence, r(A i , A j ) ≪ d |S|.Since S ⊆ A, we may utilise Theorem 2.1 to deduce that for every s ≥ 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 ϕ(S) ⊆ A i ⊆ 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 ≥ 2. Noting the fact that log |A| ≪ ε |A| ε/4 , we may choose s to be sufficiently large in terms of ε, say, s = ⌈10ǫ −1 ⌉ to obtain the desired result.Theorem 2.2 can be interpreted as a bound on the number of points of A × A that lie on the polynomial curve y = ϕ(x).Naturally, one may employ this result to further prove an incidence estimate for point sets that are of the form A × A, with A ⊆ Q satisfying |A • A| ≤ K|A| for some parameter K ≥ 1, and finite sets of curves of the form y = ϕ(x), where ϕ ∈ Q[x] satisfies ϕ(0) = 0, but we do not pursue this here.

Properties of mixed energies
We begin this section by recording some notation.For every real number λ = 0 and every where for every finite set X ⊆ R and for every ν ∈ R, we denote ν • X = {νx | x ∈ X}.Next, given some function a : R → [0, ∞) and some finite sets A 1 , . . ., A 2s ⊆ R, we define ).Moreover, when the function a satisfies a(x) = 1 for every x ∈ R, we suppress the dependence on a, and thus, we write E s (A 1 , . . ., A 2s ) = E s,a (A 1 , . . ., A 2s ).
Our first aim in this section is to prove the following generalisation of [14, Lemma 3.2].
Lemma 3.1.Let A 1 , . . ., A 2s be a finite sets of real numbers and let a : R → [0, ∞).Then Moreover, for natural number r and for finite subsets A 1 , . . ., A r ⊆ R, we have Finally, if a(x) = 1 for every x ∈ R, then for every 1 ≤ l < s, we have Proof.We begin by focusing on proving the first inequality.Firstly, note that this inequality remains invariant under replacing the function a by a/M, for any M > 0.Moreover, since A 1 , . . ., A 2s are finite sets, we may set M = max a∈A 1 ∪•••∪A 2s a(a) + 100 in the preceding statement to ensure that 0 ≤ a(a) ≤ 1 for any a ∈ A 1 ∪ • • • ∪ A 2s .We further point out that it suffices to prove that for every ε > 0, we have For the purposes of this proof, we define, for each (ξ, R) ∈ R 2 satisfying ξ = 0 and R > 0, the quantity I(R, ξ) = [0,R] e(ξα)dα.When ξ = 0, we see that |I(R, ξ)| ≪ |ξ| −1 while I(R, 0) = R.We now define, for each 1 ≤ i ≤ 2s, the exponential sum and we let Finally, we write With this discussion in hand, it is straightforward to note that for each R > 0, we have where we have used the fact that 0 ≤ a(a) ≤ 1 for any Amalgamating these expressions with a standard application of Hölder's inequality gives us , 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 ε > 0. This follows from noting the fact that |f 1 (α)| ≤ |A| for each 0 ≤ α ≤ R along with (3.1), and then choosing R to be some sufficiently large real number.
Then we have that Moreover, if for each 1 ≤ j ≤ 2s, we have some open interval I j ⊆ R such that xy > 0 and ϕ i (x)ϕ i (y) > 0 for every x, y ∈ I j and A j ⊆ I j , then Proof.We begin by noting that ϕ s+i (x s+i ) and x s+i holds true if and only if we have for every λ = 0.Moreover, note that Hence, upon dilation by an appropriate natural number λ, we may assume that ϕ is an element of Z[x] 2s as well as that A 1 , . . ., A 2s are finite subsets of Z \ ({0} ∪ Z ϕ ).
Moreover, we can further define the logarithmic map ψ X : ̺ X (X) → [0, ∞) by writing ψ X (n) = log n for every n ∈ ̺ X (X).Note that ψ X is bijective onto its image and satisfies the fact that for every z 1 , . . ., z 2s ∈ ̺ 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 ≤ i ≤ 2s, we may partition where I i,1 , . . ., I i,r i are open, pairwise disjoint intervals such that ϕ i (x)ϕ i (y) > 0 and xy > 0 for all x, y ∈ I i,j , where 1 ≤ j ≤ r i and r i ≤ d+2.Writing A i,j = A i ∩I i,j for each 1 ≤ i ≤ 2s and 1 ≤ j ≤ r i , we see that where we have crucially used the fact that A i = ∪ 1≤j≤r i A i,j for each 1 ≤ i ≤ 2s, which itself follows from the hypothesis that A 1 , . . ., A 2s are subsets of Q \ ({0} ∪ Z ϕ ).Combining the preceding expression with (3.4) and the fact that r 1 , . . ., r 2s ≤ d + 2 then delivers (3.3).
We will now prove (3.4), and so, we define the set ) is bounded by the number of solutions to the system , where each solution is being counted with the weight a(a 1 ) . . .a(a 2s ).Consequently, letting X = ∪ 1≤i≤2s X i and σ(x) = ψ X (̺ X (x)) for every x ∈ X, we see that J s,a,ϕ (A 1 , . . ., A 2s ) ≤ E s,a (σ(X 1 , ) . . ., σ(X 2s )) ≤ E s,a (σ(X 1 )) 1/2s . . .E s,a (σ(X 2s )) 1/2s , where the last inequality follows from Lemma 3.1.By the definition of σ, we see that for each 1 ≤ i ≤ 2s, the quantity E s,a (σ(X i )) is equal to the number of solutions to the system with a 1 , . . ., a 2s ∈ A i , where each such solution is counted with the weight a(a 1 ) . . .a(a 2s ).This, in turn, equals J s,a,ϕ i (A i ) since the functions x and ϕ i (x) do not change signs as x varies in A i .Combining this with the preceding discussion, we get that J s,a,ϕ (A 1 , . . ., A 2s ) ≤ J s,a,ϕ 1 (A 1 ) 1/2s . . .J s,a,ϕ 2s (A 2s ) 1/2s , which is the desired bound.
Our final result in this section allows us to prove various other relations between the above type of mixed energies.
Then for all finite subsets A 1 , . . ., A r of Q \ ({0} ∪ Z ϕ ) and for every function (3.5) For every finite subset A of Q\({0}∪Z ϕ 1 ), for every 1 ≤ l < s and for the function a(x) = 1 for each x ∈ R, we have Proof.As before, (3.5) follows from Proposition 3.2 in a straightforward manner since J s,a,ϕ j (A i ).
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 ⊆ I, where I is some interval such that the functions ϕ 1 (x) and x do not change signs as x varies in I.As before, this allows us to construct a finite set Z ⊆ (0, ∞) such that |Z| = |A| and J t,a,ϕ 1 (A) = E t,a (Z) for each 1 ≤ t ≤ s, whereupon, a straightforward application of Lemma 3.1 delivers the estimate J s,a,ϕ consequently finishing our proof of Lemma 3.3.
It is worth noting that in the hypotheses of Proposition 3.2 and Lemma 3.3, there are no lower bounds for deg ϕ i , whenceforth, we may even choose the polynomials ϕ 1 , . . ., ϕ 2s to be constant functions.In particular, if ϕ 1 (x) = • • • = ϕ 2s (x) = 1 for every x ∈ R, then we see that J s,a,ϕ (A) = M s,a (A) for every finite set A ⊆ Q.This implies that Proposition 3.2 and Lemma 3.3 hold true when the weighted mixed energies J s,a,ϕ (A) are replaced by weighted multiplicative energies M s,a (A).

Chang's lemma for additive equations and query complexity
Let ϕ(x) be a polynomial in Z[x] of degree d, for some d ∈ N. Given finite sets A ⊆ N and V ⊆ Z and some prime p, we write where for any n ∈ N, the quantity ν p (n) denotes the largest exponent m ∈ Z such that p m divides n.Moreover, we set ν p (0) = ∞.When V = {l} for some l ∈ Z, we write A p,l = A p,V , and we define ν p (A) = {ν p (a) | a ∈ 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 , . . ., f r : [0, 1] → C, we have Our main object of study in this section would be E s,a,ϕ (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 f a,ϕ (A; α) = a∈A a(a)e(αϕ(a)) when A is a non-empty set and we set f a,ϕ (A; α) = 0 if A is an empty set.By orthogonality, we see that |f a,ϕ (A; α)| 2s dα = E s,a,ϕ (A).Proof.For ease of notation, we will write f (B; α) = f a,ϕ (B; α) for every B ⊆ Z and α ∈ R, thus suppressing the dependence on a and ϕ.We begin by restricting our analysis to the case when ϕ(0) = 0, since the additive equation Thus, let ϕ(x) = i∈I β i x i for some non-empty set I ⊆ {1, . . ., d} and for some sequence {β i } i∈I of non-zero integers.Moreover, we define Denoting r = |X|, we write the elements of X in increasing order as x 1 < • • • < x r , and we decompose the set ν p (A) as ν p (A) = U 0 ∪ . . .U r+1 , where . We note that the sets U 0 , . . ., U r+1 are pairwise disjoint and that |U r+1 | ≤ 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 ≤ i ≤ r.We now claim that for any such i there exists j = j(i) ∈ I such that for every a ∈ A p,U i , we have ν p (ϕ(a)) = ν p (β 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 (−∞, x 1 ), (x 1 , x 2 ), . . ., (x r , ∞).Now suppose that a 1 , . . ., a 2s ∈ A p,U i satisfy Our next claim is that there exist distinct k 1 , k 2 ∈ {1, . . ., 2s} such that a k 1 , a k 2 ∈ A p,n for some n ∈ U i .If this was not so, then writing k 0 to be the distinct k which minimises ν p (a k ), we have that Employing Hölder's inequality now enables us to bound the right hand side by (2s) , and subsequently, we have that Substituting this into (4.3)finishes our proof.
In order to present the next lemma, we record some further notation, and thus, given a finite set A ⊆ Z, we define the query complexity q(A) to be the minimal t ∈ N such that there are functions f 1 , . . ., f t−1 : Z → P and a fixed prime number p 1 satisfying the fact that the vectors {(ν p 1 (a), . . ., ν pt (a))} a∈A are pairwise distinct, where the prime numbers p 2 , . . ., p t are defined recursively by setting Lemma 4.2.Let s, t be natural numbers, let A ⊆ N be a finite set such that q(A) = t and let a : N → [0, ∞) be a function supported on A. Then we have that Proof.For the sake of exposition, we will write E(X) = E s,a,ϕ (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 ν p (a) is distinct for each a ∈ A. Applying Lemma 4.1 along with the orthogonality relation (4.2), we infer that Moreover, since for each n ∈ ν p (A), the set A p,n may be written as A p,n = {a n } for some unique a n ∈ A, we see that E(A p,n ) = 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 , . . ., f t−1 : Z → P and a fixed prime number p 1 satisfying the fact that the vectors {(ν p 1 (a), . . ., ν pt (a))} a∈A are pairwise distinct, where the prime numbers p 2 , . . ., p t are defined recursively by setting p i = f i−1 (ν p i−1 (a)), for each 2 ≤ i ≤ t.As before, upon applying an amalgamation of Lemma 4.1 and (4.2), we see that Note that for each n ∈ ν p 1 (A), the set A p 1 ,n satisfies q(A p 1 ,n ) ≤ 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.

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.We begin this endeavour by proving an analogue of Lemma 4.2 for J s,a,ϕ (A).
Lemma 5.2.Let p be a prime number, let ϕ be a polynomial of degree d such that ϕ(0) = 0, let I ⊆ R be some open interval such that xy > 0 and ϕ(x)ϕ(y) > 0 for every x, y ∈ I, let A ⊆ I be a finite set and let a : N → [0, ∞) be a function.Then we have Proof.For ease of exposition, we write J(A) = J s,a,ϕ (A), thus suppressing the dependence on s, a, ϕ.Moreover, we import various notation from the proof of Lemma 4.1, and so, suppose that ϕ(x) = i∈I β i x i for some {0, d} ⊆ I ⊆ {0, 1, . . ., d} and for some sequence {β i } i∈I of non-zero integers.We define the elements x 1 < • • • < x r of the set X, and the sets U 0 , . . ., U r+1 precisely as in the proof of Lemma 4.1.The only difference is that in this case, we will have that Our first claim is that for any fixed 0 ≤ i ≤ r, whenever a 1 , . . ., a 2s ∈ A p,U i satisfy a 1 . . .a s = a s+1 . . .a 2s and ϕ(a 1 ) . . .ϕ(a s ) = ϕ(a s+1 ) . . .ϕ(a 2s ), then there exist distinct k 1 , k 2 ∈ {1, . . ., 2s} such that ν p (a k 1 ) = ν p (a k 2 ).We prove this by contradiction, and so, suppose a 1 , . . ., a 2s ∈ A p,U i satisfy the above system of equations and let k and k ′ be the unique elements in {1, . . ., 2s} for which ν p (a k ) is minimal and ν p (a k ′ ) is maximal.Moreover, since a 1 , . . ., a 2s ∈ A p,U i , there exist distinct j 1 , j 2 ∈ I such that ν p (β j 1 a j 1 l ) < ν p (β j 2 a j 2 l ) < ν p (β j a j l ) for every j ∈ I \ {j 1 , j 2 } and for every 1 ≤ l ≤ 2s.This implies that where ) and ).
In the case when j 2 > j 1 , utilising the fact that a 1 . . .a s = a s+1 . . .a 2s , we get which implies that ν p (0) < ∞, thus delivering the desired contradiction.Similarly if j 1 > j 2 , then we get that which subsequently implies that ν p (0) < ∞ 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 = ∪ n∈U r+1 A p,n and |U r+1 | ≤ r ≤ (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 ∈ {0, 1, . . ., 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 , . . ., B 2s 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 ≤ (d + 1) 2 give us the desired result.
As in §4, the above lemma may be iterated to furnish the following result.
Lemma 5.3.Let ϕ be a polynomial of degree d such that ϕ(0) = 0, let I ⊆ R be some open interval such that xy > 0 and ϕ(x)ϕ(y) > 0 for every x, y ∈ I, let A ⊆ I be a finite set such that q(A) = t and let a : N → [0, ∞) be a function.Then we have We will now combine this with an averaging argument to deliver bounds for M s,a,ϕ (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 : R → R d 1 and g : R → R d 2 and a : R In particular, this counts the number of solutions to the system where each solution (x 1 , . . ., x 2s ) ∈ A 2s is being counted with weights a(x 1 ) . . .a(x 2s ).Similarly, we define Lemma 5.4.Let s, d 1 , d 2 be natural numbers and let f : R → R d 1 and g : R → R d 2 and a : R → [0, ∞) be functions.Then for any finite set A of real numbers, we have that E s,a (A; g) ≤ |sf (A) − sf (A)|E s,a (A; f, g).
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 , . . ., A r , for some r ≪ d 1, such that for each 1 ≤ j ≤ r, the set A j lies in some interval I j for which we have xy > 0 and ϕ(x)ϕ(y) > 0 for every x, y ∈ I j .Noting the remark following Lemma 3.3, we may apply Lemma 3.3 for the multiplicative energies M s,a,ϕ (A 1 ∪ • • • ∪ A r ), whereupon, we see that it suffices to bound M s,a,ϕ (A i ) for each 1 ≤ i ≤ r individually.Moreover, for each such A i , we can let f (x) = log x and g(x) = log ϕ(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.
We end this section by recording the proof of Lemma 5.4.
Proof of Lemma 5.4.Let Y = {(f (a), g(a)) | a ∈ 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 → R, we define the function for every x ∈ G. Next, for every s ≥ 2, we define the function Moreover, for any finite set X, we define the function b : G → [0, ∞) by letting b(x, y) = a(a) whenever (x, y) = (f (a), g(a)) for some a ∈ A, and we set b(x, y) = 0 for every other choice of (x, y) ∈ G. Finally, we let b ′ : G → [0, ∞) be a function defined as b ′ (x, y) = b(−x, −y) for every (x, y) ∈ G.
The above notation allows us to rewrite (5.1) as Applying Young's convolution inequality, we see that sup 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.

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 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| ≤ 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 ⊆ R be a finite set, let s ≥ 16 be an integer and let E s (A) = |A| 2s−1 /K, for some K ≥ 1.Then there exists 2 ≤ s ′ ≤ s and a finite, non-empty set U ′ ⊆ s ′ A satisfying 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 ν, δ be positive real numbers such that ν ≥ 1 and let s ≥ 4 be some even number.Moreover, suppose that A ⊆ R is a finite, non-empty set such that E s (A) ≥ |A| 2s−ν .Then we either have E s/2 (A) > |A| s−ν+δ , or there exists some non-empty subset U ′ ⊆ (s/2)A satisfying such that for every m, n ∈ N ∪ {0}, we have We are now ready to prove Theorem 6.2.
Proof of Theorem 6.2.Letting l ≥ 4 be the integer satisfying 2 l ≤ s < 2 l+1 , we may use Lemma 3.1 to deduce that where we write |A| ν−1 = K.In particular, this means that whereupon, there exists some 2 ≤ i ≤ l such that 1) .
Defining ν ′ to be the real number such that Furthermore, by the preceding discussion, we have that 1) .
We may now apply Proposition 6.3 with s = 2 i and δ = (ν − 1)/(l − 1) to deduce the existence of some finite, non-empty set U ′ ⊆ 2 i−1 A satisfying such that for every m, n ∈ N ∪ {0}, we have We obtain the desired conclusion by noting that (log s)/2 ≤ l ≤ log s and substituting |A| ν−1 = K.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 ⊆ N follows in a straightforward manner from Theorem 6.2 by considering the logarithmic map from N to [0, ∞).
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 N such that |A • X • X| ≤ K|X|, for some K ≥ 1.Then there exists a subset B ⊆ A such that |B| ≥ |A|/K and q(B) ≤ log(2K).
In our proof of the above result, we will closely follow ideas and definitions as recorded in [8,Chapter 8], and so, given r ∈ N, we define the map π i,r : Z r → Z as π i,r (x 1 , . . ., x r ) = x i for every (x 1 , . . ., x r ) ∈ Z r .We denote a finite set X ⊆ Z to be a quasicube if |X| = 2.Moreover, when r ≥ 2 and X ⊆ Z r is some finite set, then we write X to be a quasicube if π r,r (X) = {y 1 , y 2 } for some distinct y 1 , y 2 ∈ Z as well as if the sets ∈ X} are also quasicubes.Morever, a subset of a quasicube is called a binary set.
We define a set V ⊆ Z r to be an axis aligned subspace if V = X 1 × • • • × X r , where for every 1 ≤ i ≤ r, we either have X i = {x i } for some x i ∈ Z or X i = Z.Moreover, if π i,r (V ) is not a singleton for some 1 ≤ i ≤ r, then we call π 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 ≤ i ≤ r be the largest number such that π i,r is a coordinate map on V and |π 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 ⊆ N are finite sets, we note that the set P = {p prime | p divides y for some y ∈ A ∪ X}, is finite, whence, writing P = {p 1 , . . ., p r } for some r ∈ N, we define the map ψ : N → Z d such that ψ(y) = (ν p 1 (y), . . ., ν pr (y)).For the purposes of this proof, let π i : Z r → Z be the projection map defined as π i (x 1 , . . ., x r ) = x i for every 1 ≤ i ≤ r.Our hypothesis now implies that |ψ(A) + ψ(X) + ψ(X)| ≤ K|ψ(X)|, and so, writing V to be the largest binary set contained in ψ(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 ′ ⊆ A such that Noting the definition of skew-dimension and query complexity, it is relatively straightforward to show that q(A ′ ) ≤ d * (ψ(A ′ )) + 1. Combining this with the preceding discussion dispenses the desired conclusion.
We first proceed to prove the bound on the mixed multiplicative energy M s,ϕ (B), and so, just for this part of the proof, we assume that ϕ j (0) = 0 for 1 ≤ j ≤ 2s.Writing q = 10k, this allows us to employ Lemma 5.1 to see that for each 1 ≤ i ≤ r and 1 ≤ j ≤ 2s.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 D is sufficiently large in terms of C. Thus, the estimate 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 §3 respectively.
We now turn to the case of dealing with the mixed additive energy E s,ϕ (B).This, in fact, proceeds in a very similar fashion to the proof for the upper bound on M s,ϕ (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,ϕ (B) is slightly simpler than the situation where we provide estimates for M s,ϕ (B), since the former does not require utilising the upper bound (7.3) on the many-fold product sets of B 1 , . . ., B r .
Thus, we have proven Theorem 1.2 when A is a finite subset of N \ Z ϕ and ϕ ∈ (Z[x]) 2s .We now reduce the more general case when A is a finite subset of Q and ϕ ∈ (Q[x]) 2s to the aforementioned setting.As in §3, we see that upon dilating the set A appropriately, we may reduce the more general case to the setting when A ⊆ Z and ϕ ∈ (Z[x]) 2s .We now focus on the multiplicative setting of Theorem 1.2 for the latter case, and so, given some finite A ⊆ Z and some ϕ ∈ (Z[x]) 2s such that ϕ j (0) = 0 for every 1 ≤ j ≤ 2s, we define ϕ ′ to satisfy ϕ ′ j (x) = ϕ j (−x) for every x ∈ Z and 1 ≤ j ≤ 2s.Next, we write Z = (Z ϕ ∪ Z ϕ ′ ) \ {0} and we denote By Theorem 1.2, we see that A similar strategy maybe followed to resolve the case when M s,ϕ (B) is replaced by E s,ϕ (B).This concludes our proof of Theorem 1.2.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| ≥ 2. We now apply Lemma 6.1 to infer that |A (3) | ≤ K 3 |A|, whence, we may apply Lemma 6.4 to deduce the existence of a set Finally, since for each 1 ≤ i ≤ r, the set A i lies in a dilate of the set A ′ , we must have q(A i ) ≤ q(A ′ ) ≤ 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).
The proof of Theorem 2.1 follows mutatis mutandis, and we make some brief remarks concerning this below.Adapting the methods of §5 along with the results of this section, that is, following ideas from §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,a,ϕ (A), where ϕ ∈ Q[x] with ϕ(0) = 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,a,ϕ (A), when |A • A| ≤ K|A|.This finishes the proof of Theorem 2.1 when ϕ = (ϕ, . . ., ϕ) for some ϕ ∈ Q[x] with ϕ(0) = 0.The more general case may then be deduced from this special case by employing Proposition 3.2 and the ideas involved therein.

Additive and multiplicative Sidon sets
We dedicate this section to proving Theorem 1.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 ∪ 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 and so, we finish the subcase when |B 1 | ≥ |B|/3.We may proceed similarly in the case when |B 2 | ≥ |B 1 |/3 to finish the proof of additive part of Theorem 1.4.As for the multiplicative case, since ϕ(0) = 0, we see that M s,ϕ (B) ≪ s,d |B| 2s−ηs .We may now continue as in the additive case to finish our proof of Theorem 1.4.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.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 for every 1 ≤ i ≤ r, where the set A i−1 \ A i is either a B + s [1] set or a B × s [1] set.A straightforward application of Lemma 6.5 implies that |A r | ≤ 1 for some r ≪ s |A| 1−δs/s .Thus, we may partition A as where X i is a B + s,ϕ [1] set for every 1 ≤ i ≤ r 1 , the set Y j is a B × s [1] set for every 1 ≤ j ≤ r 2 , the sets X 1 , . . ., X r 1 , Y 1 , . . ., Y r 2 are pairwise disjoint and r 1 + r 2 ≪ s |A| 1−δs/s .Writing B = ∪ 1≤i≤r 1 X i , we may apply Lemma 3.1 to deduce that We may now optimise this with the inequality recorded in (9.1), which gives us that m s−d(d+1)/2 = n 2s−2 , that is, n 3s−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| ≫ N.Moreover, substituting the value of n into the preceding set of inequalities yields the bound s+2 , thus, we only focus on the additive case here.By way of the prime number theorem, we may deduce that A ′ ⊆ {1, . . ., M} for some M ≪ N(log N) 2 , whence, writing X to be the largest B + s,ϕ [1] subset of A ′ , we get that This dispenses the desired bound, and so, we conclude our proof of Proposition 1.5.
holds true whenever D is sufficiently large in terms of C d .Noting the fact that for any x ∈ R and any ϕ ∈ Z[x] satisfying 1 ≤ deg ϕ ≤ d, there are at most O d (1) solutions to x = ϕ(b) with b ∈ B, we deduce that M s,ϕ (B) ≪ s,d M s (ϕ 1 (B), . . ., ϕ 2s (B)).

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 R \ {0} such that |A| ≥ 2 and |A • B| ≤ C|B|, for some C ≥ 1.Then there exists a set S ⊆ A • B −1 with |S| ≪ C log |A| such that A ⊆ S • B.
which delivers a contradiction.Combining this claim with the orthogonality relation (4.2) and applying triangle inequality, we get that