The level of distribution of the Thue--Morse sequence

The level of distribution of a complex valued sequence b measures
the quality of distribution of b along sparse arithmetic progressions nd+a.
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 θ<1.
This result improves on the level of distribution 2/3 obtained by Müllner and the author.

As an application of our method, we show that the subsequence of the Thue--Morse sequence indexed by [n^c], where 1<c<2, is simply normal.
This result improves on the range 1<c<3/2 obtained by Müllner 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.