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The level of distribution of the Thue–Morse sequence
- Part of
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- Journal:
- Compositio Mathematica / Volume 156 / Issue 12 / December 2020
- Published online by Cambridge University Press:
- 25 January 2021, pp. 2560-2587
- Print publication:
- December 2020
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- Article
- Export citation
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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 Müllner 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. 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.
