On Some Properties of Accessible Sets

A set $D\subseteq \mathbb{N}$ is called $r$-large if every $r$-coloring of $\mathbb{N}$ admits arbitrarily long monochromatic arithmetic progressions $a,a+d,\mathrm{.}\mathrm{.}\mathrm{.},a+(k-1)d$ 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 $x_1,x_2,\mathrm{.}\mathrm{.}\mathrm{.},x_k$ 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.