A sumset version of a conjecture of Pilz

TL;DR

This paper explores a sumset variant of Pilz's conjecture, demonstrating that the sumset of scaled sets contains at least n values with odd multiplicity, extending the original problem's scope.

Contribution

It introduces a sumset version of Pilz's conjecture and proves that at least n values appear an odd number of times in this sumset.

Findings

Sumset contains at least n values with odd multiplicity

Extends Pilz's conjecture to sumsets

Provides a new perspective on additive properties of scaled sets

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 $\{a_1,2a_1,\dots,na_1\},\dots, \{a_k,2a_k,\dots,na_k\}$ (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 $\{a_1,2a_1,\dots,na_1\}+\dots+\{a_k,2a_k,\dots,na_k\}$ (considered as a multiset) at least $n$ values appear an odd number of times.