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 $D$ depends on $\theta$, 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 $2/3$ 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 $1/2$ in this subsequence. This result improves on the range $1<c<3/2$ 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).