Abstract
We prove that every additive set A with energy \(E(A)\ge |A|^3/K\) has a subset \(A'\subseteq A\) of size \(|A'|\ge (1-\varepsilon )K^{-1/2}|A|\) such that \(|A'-A'|\le O_\varepsilon (K^{4}|A'|)\). This is, essentially, the largest structured set one can get in the Balog–Szemerédi–Gowers theorem.
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1 Introduction
An additive set is a nonempty finite subset of an abelian group. The energy of an additive set A is defined to be the number E(A) of quadruples \((a_1, a_2, a_3, a_4)\in A^4\) solving the equation \(a_1+a_2=a_3+a_4\). An easy counting argument shows
where \(r_{A-A}(d)\) indicates the number of representations of d as a difference of two members of A. So the Cauchy–Schwarz inequality yields \(E(A)\ge |A|^4/|A-A|\) and, in particular, every additive set A with small difference set \(A-A\) contains a lot of energy. In the converse direction Balog and Szemerédi [2] proved that large energy implies the existence of a substantial subset whose difference set is small. After several quantitative improvements (see e.g., Gowers [3] and Balog [1]) the hitherto best version of this result was obtained by the second author [4].
Theorem 1.1
Given a real \(K\ge 1\) every additive set A with energy \(E(A)\ge |A|^3/K\) has a subset \(A'\subseteq A\) of size \(|A'|\ge \Omega (|A|/K)\) such that \(|A'-A'|\le O(K^{4}|A'|)\). \(\square \)
When investigating the question how a quantitatively optimal version of this result might read, there are two different directions one may wish to pursue. First, there is the obvious problem whether the exponent 4 can be replaced by some smaller number. Second, one may try to find “the largest” set \(A'\subseteq A\) such that \(|A'-A'|\le O_K(|A'|)\) holds. As the following example demonstrates, there is no absolute constant \(\varepsilon _\star >0\) such that \(|A'|\ge (1+\varepsilon _\star )K^{-1/2}|A|\) can be achieved in general.
Fix an arbitrary natural number n. For a very large finite abelian group G we consider the additive set
whose ambient group is \(G^n\). Obviously we have
so the real number K satisfying \(E(A)=|A|^3/K\) is roughly \(n^2\). However, every \(A'\subseteq A\) of size \(|A'|\ge (1+\varepsilon )|G|\) satisfies \(|A'-A'|\ge \varepsilon ^2 |G|^2\). Our main result implies that this is, in some sense, already the worst example. More precisely, for every fixed \(\varepsilon >0\) the Balog-Szemerédi-Gowers theorem holds with \(|A'|\ge (1-\varepsilon )K^{-1/2}|A|\). Perhaps surprisingly, we can also reproduce the best known factor \(K^4\).
Theorem 1.2
Given real numbers \(K\ge 1\), \(\varepsilon \in (0, 1/2)\), and an additive set A with energy \(E(A)\ge |A|^3/K\) there is a subset \(A'\subseteq A\) such that
Our proof has two main cases and in one of them (see Lemma 3.1 below) we even get the stronger bound \(|A'-A'|\le O_\varepsilon (K^3|A'|)\). It would be interesting to prove this in the second case as well. Using examples of the form \(A=\{x\in {\mathbb {Z}}^d:\Vert x\Vert \le R\}\) one can show that the exponent 4 cannot be replaced by any number smaller than \(\log (4)/\log (27/16)\approx 2.649\) (see [5]).
2 Preliminaries
This section discusses two auxiliary results we shall require for the proof of Theorem 1.2. The first of them is similar to [6, Lemma 6.19].
Lemma 2.1
If \(\delta , \xi \in (0, 1]\) and \(R\subseteq A^2\) denotes a binary relation on a set A such that \(|R|\ge \delta |A|^2\), then there is a set \(A'\subseteq A\) of size \(|A'|\ge \delta (1-\xi ) |A|\) which possesses the following property: For every pair \((a_1, a_2)\in A'^2\) there are at least \(2^{-7}\delta ^4\xi ^4|A|^2|A'|\) triples \((x, b, y)\in A^3\) such that \((a_1, x), (b, x), (b, y), (a_2, y)\in R\).
Proof
Set \(N(x)=\{a\in A:(a, x)\in R\}\) for every \(x\in A\). Since \(\sum _{x\in A}|N(x)|=|R|\ge \delta |A|^2\), the Cauchy–Schwarz inequality yields
Setting \(K(a, a')=\{x\in A:a, a'\in N(x)\}\) for every pair \((a, a')\in A^2\) and
a double counting argument yields
Together with (2.1) we obtain
and, hence, there exists some \(x_\star \in A\) such that the set \(A_\star =N(x_\star )\) satisfies
We shall prove that the set
has all required properties. By (2.2) we have
for which reason
meaning that \(A'\) is indeed sufficiently large. To conclude the proof we need to show
for every pair \((a_1, a_2)\in A'^2\). This follows from the fact that due to the definition of \(A'\) there are at least \(|A_\star |/2\) elements \(b\in A_\star \) such that the sets \(K(a_1, b)\) and \(K(b, a_2)\) both have at least the size \(\delta ^2\xi ^2|A|/8\). \(\square \)
Lemma 2.2
Suppose that the real numbers \(x_1, \dots , x_n\in [0, 1]\) do not vanish simultaneously. Denote their sum by S and the sum of their squares by T. For every \(\alpha \in (0, 1)\) there exists a set \(I\subseteq [n]\) such that
Proof
For reasons of symmetry we may assume \(x_1\ge \dots \ge x_n\). Set \(S_i=\sum _{j=1}^i x_j\) for every nonnegative \(i\le n\). Due to \(T\le x_1 S\) and \(x_1\le 1\) we have \(T\le S=S_n\) and thus there exists a smallest index \(k\in [n]\) satisfying \(S_k\ge \alpha T\). Notice that
Moreover \(x_1\ge T/S\) implies the existence of a largest index \(\ell \) such that \(x_\ell \ge (1-\alpha )T/(2\,S)\). Due to
we have
whence, in particular, \(\ell \ge k\). Next,
entails
Now assume for the sake of contradiction that our claim fails. Every \(i\in [k, \ell ]\) satisfies \(S_i\ge S_k\ge \alpha T\) and thus the failure of \(I=[i]\) discloses
Combined with \(ix_i\le S_i\) this entails
In view of (2.3) we are thus led to
i.e., \(2^7S^2\le 27(1-\alpha )^2\ell T\), which contradicts (2.4). \(\square \)
3 The proof of Theorem 1.2
Let us fix two real numbers \(K\ge 1\) and \(\varepsilon \in (0, 1/2)\) as well as an additive set A satisfying \(E(A)\ge |A|^3/K\). We consider the partition
![](http://media.springernature.com/lw275/springer-static/image/art%3A10.1007%2Fs00493-024-00092-5/MediaObjects/493_2024_92_Equ42_HTML.png)
defined by
According to (1.1) at least one of the cases
needs to occur and we begin by analysing the left alternative.
Lemma 3.1
If \(\sum _{d\in P}r_{A-A}(d)^2\ge \varepsilon |A|^3/(4K)\), then there exists a set \(A'\subseteq A\) of size \(|A'|\ge (1-\varepsilon )K^{-1/2}|A|\) such that \(|A'-A'|\le 2^{10}\varepsilon ^{-4} K^3|A'|\).
Proof
For every difference \(d\in P\) we set \(A_d=A\cap (A+d)\). Due to \(|A_d|=r_{A-A}(d)\) the hypothesis implies
For every pair \((x, y)\in A^2\) the set \(L(x, y)=\{d\in P:x, y\in A_d\}\) has at most the cardinality \(|L(x, y)|\le r_{A-A}(x-y)\), because every difference \(d\in L(x, y)\) corresponds to its own representation \(x-y=(x-d)-(y-d)\) of \(x-y\) as a difference of two members of A. Applying this observation to all pairs in
we obtain
Together with (3.2) this yields
and, consequently, for some element \(d(\star )\in P\) the set \(A_\star =A_{d(\star )}\) satisfies \(|A_\star ^2\cap \Xi |\le \varepsilon |A_\star |^2/4\). We contend that the set
has the required properties. As in the proof of Lemma 2.1 we obtain
so it remains to derive the required upper bound on \(|A'-A'|\).
To this end we consider an arbitrary pair \((a, a')\) of elements of \(A'\). Owing to the definition of \(A'\) there are at least \(|A_\star |/2\) elements \(x\in A_\star \) such that \((a, x)\not \in \Xi \) and \((a', x)\not \in \Xi \). For each of them we have \(a-a'=(a-x)-(a'-x)\), there are at least \(\varepsilon ^2|A|/(16K)\) pairs \((a_1, a_2)\in A^2\) solving the equation \(a-x=a_1-a_2\) and at least the same number of pairs \((a_3, a_4)\in A^2\) such that \(a'-x=a_3-a_4\). Altogether there are at least
possibilities of writing \(a-a'=(a_1-a_2)-(a_3-a_4)\) and for this reason we have
\(\square \)
We conclude the proof of Theorem 1.2 by taking care of the right case in (3.1).
Lemma 3.2
If \(\sum _{d\in Q}r_{A-A}(d)^2\ge (1-\varepsilon /4)|A|^3/K\), then there is a set \(A'\subseteq A\) of size \(|A'|\ge (1-\varepsilon )K^{-1/2}|A|\) such that \(|A'-A'|\le 2^{33}\varepsilon ^{-9}K^{4}|A'|\).
Proof
Let \(Q=\{d_1, \ldots , d_{|Q|}\}\) enumerate Q. By the definition of Q there are real numbers \(x_1, \ldots , x_{|Q|}\in [0, 1]\) such that
Owing to \(\sum _{d\in A-A} r_{A-A}(d)=|A|^2\) and the hypothesis we have
By Lemma 2.2 applied with \(\alpha =1-\varepsilon /4\) there exist an index set \(I\subseteq [|Q|]\) such that
Now we set \(Q'=\{d_i:i\in I\}\), consider the relation
and define \(\delta \in (0, 1]\) by \(|R|=\delta |A|^2\). Due to
the bounds in (3.3) imply both
By Lemma 2.1 applied to \(\xi =\varepsilon /2\) and R there exists a set \(A'\subseteq A\) of size
such that for every pair \((a_1, a_2)\in A'^2\) there are at least \(2^{-11}\varepsilon ^4\delta ^4|A|^2|A'|\) triples \((x, b, y)\in A^3\) with \((a_1, x), (b, x), (b, y), (a_2, y)\in R\). Due to the equation
this means that every difference \(a_1-a_2\in A'-A'\) has at least \(2^{-11}\varepsilon ^4\delta ^4|A|^2|A'|\) representations of the form \(q_1-q_2+q_3-q_4\) with \(q_1, q_2, q_3, q_4\in Q'\), whence
Due to \(|A'|\ge (1-\varepsilon /2)\delta |A|\ge \delta |A|/\sqrt{2}\) the result follows. \(\square \)
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Reiher, C., Schoen, T. Note on the Theorem of Balog, Szemerédi, and Gowers. Combinatorica 44, 691–698 (2024). https://doi.org/10.1007/s00493-024-00092-5
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DOI: https://doi.org/10.1007/s00493-024-00092-5