The (gowers-)balog-szemerédi Theorem: an Exposition

We briefly discuss here the Balog-Szemerédi theorem, compare its different versions and prove, following Gowers, the strongest one — also due to Gowers. 1. Discussion For a finite set A of elements of an abelian group and a group element s, by νA(s) we denote the number of representations of s as a sum of two elements of A: νA(s) = #{(a′, a′′) ∈ A× A : s = a′ + a′′}. We write 2A = {a′ + a′′ : a′, a′′ ∈ A}, the set of all elements s with νA(s) > 0. Our motivation will be clear from the following simple lemma. Lemma 1. Suppose that |2A| ≤ C|A| with some real C > 0. Then (i) there are at least |A|/2 elements s ∈ 2A with νA(s) ≥ |A|/(2C); (ii) if W is the set of all pairs (a′, a′′) ∈ A×A with νA(a+a) ≥ |A|/(2C), then |W | ≥ |A|2/(4C) and furthermore there are at most 2C|A| distinct sums of the form a′ + a′′ with (a′, a′′) ∈ W ; (iii) the number of solutions of the equation a1 + a2 = a3 + a4 in the variables a1, a2, a3, a4 ∈ A is at least |A|3/C. Proof. Write S := {s ∈ 2A : νA(s) ≥ |A|/(2C)}. Then |A| = ∑