Complexity of Hartman sequences

Abstract

Let T:x↦x+g be an ergodic translation on the compact group C and M⊆C a continuity set, i.e. a subset with topological boundary of Haar measure 0. An infinite binary sequence a:ℤ↦{0,1} defined by a(k)=1 if T k (0 C )∈M and a(k)=0 otherwise, is called a Hartman sequence. This paper studies the growth rate of P a (n), where P a (n) denotes the number of binary words of length n∈ℕ occurring in a. The growth rate is always subexponential and this result is optimal. If T is an ergodic translation x↦x+α (α=(α 1 ,...,α s )) on 𝕋 s and M is a box with side lengths ρ j not equal α j ℤ+ℤ for all j=1,...,s, we show that lim n P a (n)/n s =2 s ∏ j=1 s ρ j s-1 .