More differences than multiple sums

TL;DR

This paper investigates the relationship between the size of difference sets and multiple sum sets, demonstrating the existence of sets where the sum set is significantly smaller than the difference set.

Contribution

It introduces sets with a smaller sum set size compared to the difference set, showing that the sum set can be substantially smaller than the difference set raised to a power less than one.

Findings

Existence of sets with |kA| < |A-A|^{a_k} for some a_k<1

Demonstrates that difference sets can be larger than multiple sum sets in certain cases

Provides new insights into the structure of sum and difference sets in additive combinatorics.

Abstract

We compare the size of the difference set $A-A$ to that of the set $kA$ of $k$-fold sums. We show the existence of sets such that $|kA| < |A-A|^{a_k}$ with $a_k<1$.