Gaps problems and frequencies of patches in cut and project sets

TL;DR

This paper explores the relationship between gaps in Diophantine approximation and the frequency spectrum of patches in cut and project sets, providing bounds and conditions for frequency behavior.

Contribution

It establishes new bounds on the number of patch frequencies in cut and project sets and links these to gaps problems in Diophantine approximation.

Findings

Bounds on the number of patch frequencies depending on the set structure

Almost always fewer than a power of log r frequencies

Existence of sets with bounded frequency counts as r increases

Abstract

We establish a connection between gaps problems in Diophantine approximation and the frequency spectrum of patches in cut and project sets with special windows. Our theorems provide bounds for the number of distinct frequencies of patches of size r, which depend on the precise cut and project sets being used, and which are almost always less than a power of log r. Furthermore, for a substantial collection of cut and project sets we show that the number of frequencies of patches of size r remains bounded as r tends to infinity. The latter result applies to a collection of cut and project sets of full Hausdorff dimension.