The Erd\H{o}s distinct subset sums problem in a modular setting

TL;DR

This paper investigates the Erdős distinct subset sums problem in a modular setting, establishing bounds on the maximum element of sumset-distinct sets modulo a specific number and characterizing their structure for small parameters.

Contribution

It proves a lower bound on the maximum element of sumset-distinct sets modulo 2^n+t and characterizes their structure for small t, showing the constant 1/3 is optimal.

Findings

Maximum element of sumset-distinct sets is at least (1/3 - o(1)) times N.

The constant 1/3 is proven to be best possible.

Structural characterization of sumset-distinct sets for small t.

Abstract

We prove the following variant of the Erd\H{o}s distinct subset sums problem. Given $t \ge 0$ and sufficiently large $n$, every $n$-element set $A$ whose subset sums are distinct modulo $N=2^n+t$ satisfies $$\max A \ge \Big(\frac{1}{3}-o(1)\Big)N.$$ Furthermore, we provide examples showing that the constant $\frac 13$ is best possible. For small values of $t$, we characterise the structure of all sumset-distinct sets modulo $N=2^n+t$ of cardinality $n$.