Arithmetical structure of sumset intersections

Diego Marques; Melvyn B. Nathanson

paper

Arithmetical structure of sumset intersections

arXiv:2603.14510

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 $A = ⋂_{q = 1}^{\infty} A_{q}$ . Then $h A \subseteq ⋂_{q = 1}^{\infty} h A_{q}$ for all $h \geq 1$ . Let $H (A_{q}) = {h \geq 1 : h A = ⋂_{q = 1}^{\infty} h A_{q}}$ . The arithmetical structure of the sets $H (A_{q})$ is unknown. It is proved that for every $h_{0} \geq 2$ there exist sequences $(A_{q})_{q = 1}^{\infty}$ such that ${1, \dots, h_{0} - 1} \subseteq H (A_{q})$ but $h_{0} \in / H (A_{q})$ and also that there exist sequences $(A_{q})_{q = 1}^{\infty}$ such that ${1, h_{0}} \subseteq H (A_{q})$ but ${2, 3, \dots, h_{0} - 1} \cap H (A_{q}) = \emptyset$ .

7 pages