Intersections of sumsets in additive number theory

TL;DR

The paper investigates when taking an $h$-fold sumset commutes with infinite intersections under decreasing families of sets across additive structures. It proves $hA = \bigcap_{q=1}^{\infty} hA_q$ for all $h$ in the nonnegative integers under $A = \bigcap_{q=1}^{\infty} A_q$, and characterizes when this holds in terms of $A$ being a basis of order $h_0$ for $\mathbb{Z}$. In the setting of locally compact abelian groups, the equality holds for all $h$ under compactness, with a Haar-measure analogue giving $\mu(hA) = \lim_{q\to\infty} \mu(hA_q)$. The work also outlines open questions for additive semigroups regarding the exact set of $h$ for which equality holds and whether this property propagates to adjacent values, highlighting a broader framework for sumset–intersection behavior.

Abstract

Let $A$ be a subset of an additive abelian semigroup 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 the semigroup and let $A = \bigcap_{q=1}^{\infty} A_q$. When does one have \[ hA = \bigcap_{q=1}^{\infty} hA_q \] for some or all $h \geq 2$?

Theorems & Definitions (12)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof
• Theorem 5
• proof