Uniform distribution modulo 2 by an endomorphism of the circle

Abstract

If T is an endomorphism of degree 2, then we know that the sequence $y_n ∈ {0.1}$ defined by $y_n(x) = χ_{[1/2,1]}(T^nx)$ is uniformly distributed by the classical Borel``s Theorem on normal numbers. In this article, we are interested in the uniform distribution of the sequence $y_n ∈ {0.1}$ defined by $y_n(X) = ∑^{n-1}_{k=0} χ_E(T^kx) (mod 2)$. We show that if E is an interval with binary fraction end points, then the sequence is uniformly distributed in $L^2$ sense. In particular if E = [1/4,3/4], then the sequence is uniformly distributed almost everywhere.