Note on the theorem of Balog, Szemerédi, and Gowers (Q6548029)

In this short and well-written note, the authors give a full proof of an improved explicit version of the Balog - Szemerédi - Gowers Theorem see \textit{W. T. Gowers} [Geom. Funct. Anal. 8, No. 3, 529--551 (1998; Zbl 0907.11005)]. The main Theorem reads as follows. \par ``Given real numbers \(K\ge1\), \(\varepsilon\in(0,1/2)\) and a finite subset \(A\) of an abelian group with energy \(E(A) \ge \vert A\vert ^3 /K\) there is a subset \(A'\subset A\) such that \(\vert A'\vert \ge (1 - \varepsilon)K^{-1/2} \vert A\vert \) and \(\vert A'-A\vert \le 2^{33}\varepsilon^{-9} K^4 \vert A'\vert \).''\par In some cases, the factor \(K^4\) may even be reduced to \(K^3\). The proof goes by splitting \(A-A'\) into a set \(P\) where \(r_{A-A'}\) is large and \(Q\) where it is small and analysing differently the case when \(P\) is large (in a weighted sense) and the one when \(Q\) is large.