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 theta < 1. This result improves on the level of distribution 2/3 obtained by Mullner 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 Mullner 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.