RESTRICTED ADDITION AND SOME DEVELOPMENTS OF THE ERDŐS–GINZBURG–ZIV THEOREM

Abstract

Thanks to Szemerédi's theorem on sets with no long arithmetic progressions, an elementary trick is used here to show that for a given positive integer $h$ and a given set $U$ of residue classes modulo $n$ with positive density, there exists a dense subset $V$ of $U$; that is, $U\smallsetminus V$ is very small, such that the sumset $hV$ is included in the restricted sumset $h\times U$. The next step is to obtain information on the structure of $V$ from Kneser's theorem on the sum of sets in an abelian group, and to use this for studying the structure of $U$ itself. Finally, this idea is used in the paper to derive some new values of a function related to the Erdős–Ginzburg–Ziv theorem.