Sums of dilates

TL;DR

This paper establishes sharp lower bounds and improved upper bounds for the size of sumsets formed by dilates of a set, advancing understanding in additive combinatorics.

Contribution

It provides the first asymptotically sharp lower bounds and stronger upper bounds for sumsets of dilates, generalizing classical inequalities.

Findings

Sharp lower bounds for sumsets of dilates

Enhanced upper bounds surpassing Plunnecke's inequality

Applicable to arbitrary integer sets and coefficients

Abstract

The lambda-dilate of a set A is lambda*A={lambda a : a \in A}. We give an asymptotically sharp lower bound on the size of sumsets of the form lambda_1*A+...+lambda_k*A for arbitrary integers lambda_1,...,lambda_k and integer sets A. We also establish an upper bound for such sums, which is similar to, but often stronger than Plunnecke's inequality.