Normality of the Thue--Morse sequence along Piatetski-Shapiro 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 $({\epsilon }_0,\dots ,{\epsilon }_{T-1})\in \{0,1{\}}^T$ occurs as a contiguous subsequence of the sequence $n\mapsto \mathbf{t}\left(\lfloor n^c\rfloor \right)$ with asymptotic frequency $2^{-T}$.

13 pages, published in The Quarterly Journal of Mathematics