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 language | English |
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Article number | rnab056 |
Journal | International Mathematics Research Notices |
Early online date | 23 Apr 2021 |
DOIs | |
Publication status | E-pub ahead of print - 23 Apr 2021 |