Normality of the Thue--Morse sequence along Piatetski-Shapiro sequences, II (1511.01671v2)

Abstract: We prove that the Thue--Morse sequence $\mathbf t$ along subsequences indexed by $\lfloor n^c\rfloor$ is normal, where $1<c<3/2$. That is, for $c$ in this range and for each $\omega\in{0,1}^L$, where $L\geq 1$, the set of occurrences of $\omega$ as a subword (contiguous finite subsequence) of the sequence $n\mapsto \mathbf t_{\lfloor n^c\rfloor}$ has asymptotic density $2^{-L}$. This is an improvement over a recent result by the second author, which handles the case $1<c<4/3$. In particular, this result shows that for $1<c<3/2$ the sequence $n\mapsto \mathbf t_{\lfloor n^c\rfloor}$ attains both of its values with asymptotic density $1/2$, which improves on the bound $c<1.4$ obtained by Mauduit and Rivat (who obtained this bound in the more general setting of $q$-multiplicative functions, however) and on the bound $c\leq 1.42$ obtained by the second author. In the course of proving the main theorem, we show that $2/3$ is an admissible level of distribution for the Thue--Morse sequence, that is, it satisfies a Bombieri--Vinogradov type theorem for each exponent $\eta<2/3$. This improves on a result by Fouvry and Mauduit, who obtained the exponent $0.5924$. Moreover, the underlying theorem implies that every finite word $\omega\in{0,1}^L$ is contained as an arithmetic subsequence of $\mathbf t$.