Arithmetical structure of sumset intersections

TL;DR

This paper investigates the arithmetical structure of sumset intersections of decreasing integer sets, revealing complex behaviors and constructing sequences with specific properties regarding their sumset intersection characteristics.

Contribution

It introduces new constructions of decreasing integer sets demonstrating diverse and intricate sumset intersection properties, advancing understanding of their arithmetical structure.

Findings

Existence of sequences with specific sumset intersection properties for given h_0.

Construction of sequences where certain h-values are in or out of the intersection set.

Demonstration of complex arithmetical structures in sumset intersections.

Abstract

The $h$-fold sumset of a set $A$ of integers is the set of all sums of $h$ not necessarily distinct elements of $A$. Let $(A_q)_{q=1}^{\infty}$ be a strictly decreasing sequence of sets of integers and let $A = \bigcap_{q=1}^{\infty} A_q$. Then $hA \subseteq \bigcap_{q=1}^{\infty} hA_q$ for all $h \geq 1$. Let $\mathcal{H}(A_q) = \{h \geq 1: hA = \bigcap_{q=1}^{\infty} hA_q\}$. The arithmetical structure of the sets $\mathcal{H}(A_q)$ is unknown. It is proved that for every $h_0 \geq 2$ there exist sequences $(A_q)_{q=1}^{\infty}$ such that $\{1,\ldots, h_0-1\} \subseteq \mathcal{H}(A_q)$ but $h_0 \notin \mathcal{H}(A_q)$ and also that there exist sequences $(A_q)_{q=1}^{\infty}$ such that $\{1, h_0 \} \subseteq \mathcal{H}(A_q)$ but $\{2,3, \ldots, h_0-1\} \cap \mathcal{H}(A_q) = \emptyset$.