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We prove that for   $1&lt;c&lt;4/3$   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   $2^{-T}$  .

NORMALITY OF THE THUE–MORSE SEQUENCE ALONG PIATETSKI-SHAPIRO SEQUENCES