Intersections of shifted sets

TL;DR

This paper investigates how shifting sets of natural numbers by elements of other sets affects their intersections, revealing new properties related to the size of the sets and recurrence behaviors.

Contribution

It establishes intersection properties based on the relative asymptotic sizes of the sets and links these to recurrence sets containing large finite set distance sets.

Findings

Sets with growth rate o(n^{k/(k-1)}) have recurrence sets containing large finite set distances.

Intersection properties depend on the relative asymptotic sizes of the involved sets.

Abstract

We consider shifts of a set $A\subseteq\mathbb{N}$ by elements from another set $B\subseteq\mathbb{N}$, and prove intersection properties according to the relative asymptotic size of $A$ and $B$. A consequence of our main theorem is the following: If $A=\{a_n\}$ is such that $a_n=o(n^{k/k-1})$, then the $k$-recurrence set $R_k(A)=\{x\mid |A\cap(A+x)|\ge k\}$ contains the distance sets of arbitrarily large finite sets.