On a theorem of Shkredov (Q2885115)

Let \(G\) be an abelian group and let \(A\) be a finite subset of \(G\). For any finite set \(L \subset G\), let \(Span(L)\) denote the set of all the linear combinations of the elements of \(L\) with coefficients from the set \(\{-1, 0, 1\}\). In this paper the author improves a result of \textit{I. D. Skhredov} [Mat. Zametki, 84, No. 6, 927--947 (2008; Zbl 1219.11019)] by proving that if \(A\) has additive energy at least \(c|A|^{3}\) then there exists a set \(L \subset A\) with \(|L| = O(c^{-1}\log |A|)\) such that \(|A \cap Span(L)| = \Omega(c^{1/3}|A|)\). He also proves that if \(|A + A| \leq K|A|\), then there is a set \(L \subset A\) with \(|L| = O(K\log |A|)\) such that \(A \subset Span(L)\). The proofs use some tools from harmonic analysis, for example the Rudin inequality.