A sumset version of a conjecture of Pilz

János Nagy; Péter Pál Pach

paper

A sumset version of a conjecture of Pilz

arXiv:2409.15075

Abstract

Pilz's conjecture states that for any finite set $A = {a_{1}, a_{2}, \dots, a_{k}}$ of positive integers and positive integer $n$ in the union of the sets ${a_{1}, 2 a_{1}, \dots, n a_{1}}, \dots, {a_{k}, 2 a_{k}, \dots, n a_{k}}$ (considered as a multiset) at least $n$ values appear an odd number of times. In this short note we consider a variant of this problem. Namely, we show that in the sumset ${a_{1}, 2 a_{1}, \dots, n a_{1}} + \dots + {a_{k}, 2 a_{k}, \dots, n a_{k}}$ (considered as a multiset) at least $n$ values appear an odd number of times.