On dilates sums

TL;DR

This paper investigates bounds on the size of sumsets involving dilates of a finite set of integers, providing new lower bounds and extending previous results in additive combinatorics.

Contribution

It establishes a new lower bound for the sumset involving two dilates, generalizing prior bounds and applying to large sets with specific size conditions.

Findings

Derived a new lower bound for |2A + kA| when |A| > 8k^k

Extended known bounds to more general sumsets involving dilates

Identified cases where the bound is tight, such as arithmetic progressions

Abstract

Let $A$ be a finite nonempty set of integers. An asymptotic estimate of several dilates sum size was obtained by Bukh. The unique known exact bound concerns the sum $|A+k\cdot A|,$ where $k$ is a prime and $|A|$ is large. In its full generality, this bound is due to Cilleruelo, Serra and the first author. Let $k$ be an odd prime and assume that $|A|>8k^{k}.$ A corollary to our main result states that $|2\cdot A+k\cdot A|\ge (k+2)|A|-k^2-k+2.$ Notice that $|2\cdot P+k\cdot P|=(k+2)|P|-2k,$ if $P$ is an arithmetic progression.