Equidistribution results for self-similar measures

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Abstract

A well-known theorem due to Koksma states that for Lebesgue almost every x>1 the sequence (xn)∞n=1 is uniformly distributed modulo one. In this paper, we give sufficient conditions for an analogue of this theorem to hold for a self-similar measure. Our approach applies more generally to sequences of the form (fn(x))∞n=1 where (fn)∞n=1 is a sequence of sufficiently smooth real-valued functions satisfying some nonlinearity conditions. As a corollary of our main result, we show that if C is equal to the middle 3rd Cantor set and t≥1⁠, then with respect to the natural measure on C+t, for almost every x⁠, the sequence (xn)∞n=1 is uniformly distributed modulo one.
Original languageEnglish
Article numberrnab056
JournalInternational Mathematics Research Notices
Early online date23 Apr 2021
DOIs
Publication statusE-pub ahead of print - 23 Apr 2021

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