Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Arithmetical structure of sumset intersections

Diego Marques, Melvyn B. Nathanson

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 $\{2,3, \ldots, h_0-1\} \cap \mathcal{H}(A_q) = \emptyset$.

Arithmetical structure of sumset intersections

Abstract

The -fold sumset of a set of integers is the set of all sums of not necessarily distinct elements of . Let be a strictly decreasing sequence of sets of integers and let . Then for all . Let . The arithmetical structure of the sets is unknown. It is proved that for every there exist sequences such that but and also that there exist sequences such that but .
Paper Structure (2 sections, 7 theorems, 50 equations)

This paper contains 2 sections, 7 theorems, 50 equations.

Table of Contents

  1. A sumset intersection problem
  2. Arithmetical structure

Key Result

Theorem 1

Let $h \in \mathbf N$. Let $X$ be an additive abelian semigroup such that $r_{X ,h}(x) < \infty$ for all $x \in X$. Let $A$ be a subset of $X$ and let $(A_q)_{q=1}^{\infty}$ be an asymptotically strictly decreasing sequence of subsets of $X$ such that $A = \bigcap_{q=1}^{\infty} A_q$. Then and $h \in \mathcal{H}(A_q)$. If $r_{X ,h}(x) < \infty$ for all $x \in X$ and $h \geq 2$, then for all $h

Theorems & Definitions (11)

  • Theorem 1: Nathanson nath26Aus
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • Corollary 1
  • proof
  • ...and 1 more