Thue-Morse at Multiples of an Integer

TL;DR

This paper investigates the properties of the Thue-Morse sequence at multiples of an integer, proving bounds on the minimal index with a specific binary property and exploring generalizations to other bases.

Contribution

It establishes bounds on the minimal index n such that t_{kn}=1 with limited Hamming weight, and characterizes cases where minimal n equals k through k+4, extending to arbitrary bases.

Findings

Existence of n<=k+4 with t_{kn}=1 and Hamming weight <=3

Characterization of k where minimal n equals k to k+4

Conjectures for generalized bases s_b

Abstract

Let (t_n) be the classical Thue-Morse sequence defined by t_n = s_2(n) (mod 2), where s_2 is the sum of the bits in the binary representation of n. It is well known that for any integer k>=1 the frequency of the letter "1" in the subsequence t_0, t_k, t_{2k}, ... is asymptotically 1/2. Here we prove that for any k there is a n<=k+4 such that t_{kn}=1. Moreover, we show that n can be chosen to have Hamming weight <=3. This is best in a twofold sense. First, there are infinitely many k such that t_{kn}=1 implies that n has Hamming weight >=3. Second, we characterize all k where the minimal n equals k, k+1, k+2, k+3, or k+4. Finally, we present some results and conjectures for the generalized problem, where s_2 is replaced by s_b for an arbitrary base b>=2.