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AbstractWe 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 sizer, which depend on the precise cut and project sets being used, and which are almost always less than a power of logr. Furthermore, for a substantial collection of cut and project sets we show that the number of frequencies of patches ofdoi:10.1017/s0305004116000128 fatcat:ou6yj4pilrefvosfh4pfqk5y5m