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 $| defines a normal sequence, that is, each finite sequence |$(\varepsilon _0,\ldots ,\varepsilon _{T-1})\in \{0,1\}^T$| occurs as a contiguous subsequence of the sequence |$n\mapsto \mathbf t(\lfloor n^c\rfloor )$| with asymptotic frequency |$2^{-T}$|⁠.

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