Intersections of sumsets in additive number theory

TL;DR

This paper investigates conditions under which the intersection of sumsets equals the sumset of the intersection in additive number theory, focusing on decreasing sequences of sets within abelian semigroups.

Contribution

It provides new insights into the relationship between sumsets and intersections for decreasing sequences of sets in additive structures.

Findings

Established criteria for when hA equals the intersection of hA_q for decreasing sequences

Extended understanding of sumset behavior in additive abelian semigroups

Identified conditions affecting the equality of sumset intersections

Abstract

Let $A$ be a subset of an additive abelian semigroup $S$ and let $hA$ be the $h$-fold sumset of $A$. The following question is considered: Let $(A_q)_{q=1}^{\infty}$ be a strictly decreasing sequence of sets in $S$ and let $A = \bigcap_{q=1}^{\infty} A_q$. When does one have \[ hA = \bigcap_{q=1}^{\infty} hA_q \] for some or all $h \geq 2$?