2020/02/26 by Baker, Simon
#Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2002.11607
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 self-similar measures. 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 a nonlinearity assumption. As a corollary of our main result, we show that if C is equal to the middle third Cantor set and t≥ 1, then with respect to the Cantor-Lebesgue measure on C+t the sequence (xn)n=1∞ is uniformly distributed for almost every x.