The level of distribution of the Thue--Morse sequence

The level of distribution of a complex valued sequence $b$ measures “how well $b$ behaves” on arithmetic progressions $nd+a$. Determining whether $\theta$ is a level of distribution for $b$ involves summing a certain error over $d\leq D$,where$D$dependson$\theta$, this error is given by comparing a finite sum of $b$ along $nd+a$ and the expected value of the sum. We prove that the Thue–Morse sequence has level of distribution $1$, which is essentially best possible. More precisely, this sequence gives one of the first nontrivial examples of a sequence satisfying a Bombieri–Vinogradov type theorem for each exponent $\theta <1$. In particular, this result improves on the level of distribution $2\mathrm{/}3$ obtained by M"ullner and the author. As an application of our method, we show that the subsequence of the Thue–Morse sequence indexed by $\lfloor n^c\rfloor$, where $1<c<2$, is simply normal. That is, each of the two symbols appears with asymptotic frequency $1\mathrm{/}2$ in this subsequence. This result improves on the range $1<c<3\mathrm{/}2$ obtained by M"ullner and the author and closes the gap that appeared when Mauduit and Rivat proved (in particular) that the Thue–Morse sequence along the squares is simply normal. In the proofs, we reduce both problems to an estimate of a certain Gowers uniformity norm of the Thue–Morse sequence similar to that given by Konieczny (2017).