Gaps problems and frequencies of patches in cut and project sets

摘要

We establish a connection between gaps problems in Diophantine approximationand the frequency spectrum of patches in cut and project sets with specialwindows. Our theorems provide bounds for the number of distinct frequencies ofpatches 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 asubstantial collection of cut and project sets we show that the number offrequencies of patches of size r remains bounded as r tends to infinity. Thelatter result applies to a collection of cut and project sets of full Hausdorffdimension.