Additive dimension and a theorem of Sanders (Q2788674)

A finite subset \(A\) of an abelian group is called dissociated if every sum formed by the elemnts of \(A\) are different. The size of the largest dissociated subset of a set \(B\) is called the dimension of \(B\). \textit{T. Sanders} proved in [ Can. Math. Bull. 56, No. 2, 412--423 (2013; Zbl 1310.11013)] that if \(A\) and \(B\) are finite subsets such that the cardinality of the sumset \(A + B\) is at most \(K|A|\) then the dimension of \(B\) is \(\ll K\log |A|\).NEWLINENEWLINEIn this paper the authors give another proof of this result by using combinatorial tools. When \(K\) is not too large they obtain a better estimation for the dimension of \(B\). \textit{I. D. Shkredov} and \textit{S. Yekhanin} proved in [J. Comb. Theory, Ser A 118, No. 3, 1086--1093 (2011; Zbl 1238.05281)] that if \(A\) and \(B\) are finite subsets of an abelian group with aditive energy \(E(A, B)\) is at least \(|A||B|^{2}/K\), then there exists a subset \(B_{1}\) of \(B\) such that the dimension of \(B_{1}\) is \(\ll K\log |A|\) and \(E(A, B_{1}) \geq 2^{-5}E(A, B)\). In this paper the authors improve and generalize this result to other energies. In the proofs they use Fourier analysis. As an application they improve the upper bound in a problem due to Konyagin. They also prove a result of Bateman and Katz to general abelian group.