On Some Properties of Accessible Sets

TL;DR

This paper investigates properties of accessible sets in number theory, showing that 2-accessibility constrains the growth of gaps in such sets and establishing their equivalence to topological recurrence.

Contribution

It demonstrates that 2-accessible sets have gaps that grow at most exponentially and proves the equivalence between accessibility and topological recurrence.

Findings

Gaps in 2-accessible sets cannot grow faster than exponentially.

Accessibility is equivalent to topological recurrence.

Provides bounds on the growth of gaps in accessible sets.

Abstract

A set $D \subseteq \mathbb{N}$ is called $r$-large if every $r$-coloring of $\mathbb{N}$ admits arbitrarily long monochromatic arithmetic progressions $a,a+d,...,a+(k-1)d$ with gap $d \in D$. Closely related to largeness is accessibility; a set $D \subseteq \mathbb{N}$ is called $r$-accessible if every $r$-coloring of $\mathbb{N}$ admits arbitrarily long monochromatic sequences $x_1,x_2,...,x_k$ with $x_{i+1}-x_{i} \in D$. It is known that if $D \subseteq \mathbb{N}$ is $2$-large, then the gaps between elements in $D$ cannot grow exponentially. In this paper, we show that if $D$ is $2$-accessible, then the gaps between elements in $D$ cannot grow much faster than exponentially. Additionally, we show that the notion of accessibility is equivalent to that of topological recurrence.