A linear bound on the dimension in Green-Ruzsa's theorem
Let \(G\) be any abelian group. If \(H\) is a subgroup of \(G\) and \[ P=\{x_0+l_1x_1+\cdots+l_dx_d:\;l_i\in\{-L_i,\ldots,0,\ldots,L_i\}\;\text{for}\;i=1,\ldots,l\} \] is a generalized arithmetic progression of dimension \(d\) and size \(\prod_{i=1}^d(2L_i+1)\), then we call \(P+H=\{x+h:\;x\in P\;\text{and}\;h\in H\}\) a coset progression. Let \(A\) be a nonempty subset of \(G\) with \(|A+A|\leq K|A|\), where \(K\) is a positive constant. \textit{B. Green} and \textit{I. Z. Ruzsa} [J. Lond. Math. Soc., II. Ser. 75, No. 1, 163--175 (2007; Zbl 1133.11058)] showed that \(A\) is contained in a coset progression \(P+H\) of dimension \(d(P+H)\leq CK^4\log(K+2)\) and \(\text{size}(P+H)\leq\text{exp}(CK^4\log^2(K+2))|A|\), where \(C\) is a positive constant. Motivated by \textit{M.-C. Chang}'s work [Duke Math. J. 113, No. 3, 399--419 (2002; Zbl 1035.11048)] on a bound in Freiman's theorem, the authors proved that either there is a proper coset progression \(P+H\) of dimension \(d(P+H)\leq 2|K|\) and \(\text{size}(P+H)\leq \text{exp}(CK^4\log^2(K+2))|A|\) such that \(A\subseteq (P+H)\), or \(A\) is fully contained in at most \(CK^3\log^2K\) cosets (of a subgroup of \(G\)), whose total cardinality is bounded by \(\text{exp}(CK^4\log^2(K+2))|A|\).
scientific article; zbMATH DE number 6138268
scientific article