On a sumset problem for dilated integer sets

Abstract

Let A be a non-empty finite set of integers. For any integer k, let [Formula: see text]. One of the important problems in additive combinatorics is to find the optimal lower bound for the cardinality of the sumset [Formula: see text]. Cilleruelo, Silva, and Vinuesa conjectured that, if k is a positive integer and A is a finite set of integers with sufficiently large cardinality, then [Formula: see text]. For some values of k this conjecture is already confirmed. In this article, under certain restrictions on set A, we confirm this conjecture for all positive integers k except for those of the form [Formula: see text] or [Formula: see text], where p is an odd prime number and [Formula: see text] for [Formula: see text].