Normality of the Thue--Morse sequence along Piatetski-Shapiro sequences

TL;DR

This paper proves that the subsequence of the Thue--Morse sequence, indexed by Piatetski-Shapiro sequences with c between 1 and 4/3, is normal, meaning all finite patterns occur with equal asymptotic frequency.

Contribution

It establishes the normality of Thue--Morse subsequences along Piatetski-Shapiro sequences within a specific range of c, a novel result in sequence normality.

Findings

Subsequence is normal for 1 < c < 4/3.

Each finite pattern appears with asymptotic frequency 2^{-T}.

Extends understanding of normality in automatic sequences.

Abstract

We prove that for $1<c<4/3$ the subsequence of the Thue--Morse sequence $\mathbf t$ indexed by $\lfloor n^c\rfloor$ defines a normal sequence, that is, each finite sequence $(\varepsilon_0,\ldots,\varepsilon_{T-1})\in \{0,1\}^T$ occurs as a contiguous subsequence of the sequence $n\mapsto \mathbf t\left(\lfloor n^c\rfloor\right)$ with asymptotic frequency $2^{-T}$.