Two divisibility problems on subset sums

TL;DR

This paper investigates divisibility properties of subset sums in sets of integers, providing asymptotic results and structural insights for two related problems involving multiples and free-sequences.

Contribution

It offers new asymptotic bounds for the divisibility problem and refines existing results on the structure of sets with multiple-free subset sums.

Findings

Established asymptotic estimates for the first divisibility problem.

Improved bounds and structural characterizations for sets with multiple-free subset sums.

Abstract

We consider two problems regarding some divisibility properties of the subset sums of a set $A\subseteq \{1, 2, \ldots ,n\}$. At the beginning, we study the cardinality of $A$ which has the following property: For every $d\le n$ there is a non empty set $A_d\subseteq A$ such that the sum of the elements of $A_d$ is a multiple of $d$. Next, we turn our attention to another problem: If all subset sums of $A$ form a multiple free-sequence, what can we say about the structure of $A$? We give some asymptotics for the first problem and improve some already existing results for the second one.