## Abstract

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

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.

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.

Original language | English |
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Pages (from-to) | 2560-2587 |

Number of pages | 28 |

Journal | Compositio mathematica |

Volume | 156 |

Issue number | 12 |

DOIs | |

Publication status | Published - 25 Jan 2021 |

### Bibliographical note

Publisher Copyright:© The Author(s) 2021.

## Keywords

- Thue--Morse sequence
- level of distribution
- Bombieri--Vinogradov Theorem
- Thue-Morse sequence
- normal sequence
- Bombieri-Vinogradov theorem
- Elliott-Halberstam conjecture
- Piatetski-Shapiro sequence
- Gel'fond problem
- arithmetic progression