Gaps problems and frequencies of patches in cut and project sets

TY - JOUR

T1 - Gaps problems and frequencies of patches in cut and project sets

AU - Haynes, Alan

AU - Koivusalo, Henna Lotta Louisa

AU - Walton, Jamie

AU - Sadun, Lorenzo

PY - 2016/7

Y1 - 2016/7

N2 - 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.

AB - 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.

U2 - 10.1017/S0305004116000128

DO - 10.1017/S0305004116000128

M3 - Article

SN - 1469-8064

VL - 161

SP - 65

EP - 85

JO - Mathematical Proceedings of the Cambridge Philosophical Society

JF - Mathematical Proceedings of the Cambridge Philosophical Society

IS - 1

ER -