(k+1)-sums versus k-sums

TL;DR

This paper establishes an upper bound on the ratio of the number of (k+1)-sums to k-sums for a set of integers, showing it is maximized by geometric progressions, thus answering a question posed by Ruzsa.

Contribution

It proves a tight inequality relating (k+1)-sums and k-sums of integer sets, characterizing when the maximum ratio is achieved.

Findings

The ratio of (k+1)-sums to k-sums is at most (|A|-k)/(k+1).

Equality holds for geometric progressions.

The result confirms a conjecture of Ruzsa.

Abstract

A $k$-sum of a set $A\subseteq \mathbb{Z}$ is an integer that may be expressed as a sum of $k$ distinct elements of $A$. How large can the ratio of the number of $(k+1)$-sums to the number of $k$-sums be? Writing $k\wedge A$ for the set of $k$-sums of $A$ we prove that \[ \frac{|(k+1)\wedge A|}{|k\wedge A|}\, \le \, \frac{|A|-k}{k+1} \] whenever $|A|\ge (k^{2}+7k)/2$. The inequality is tight -- the above ratio being attained when $A$ is a geometric progression. This answers a question of Ruzsa.