## Rarified sums of the Thue-Morse sequence

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- by Michael Drmota and Mariusz Skałba PDF
- Trans. Amer. Math. Soc.
**352**(2000), 609-642 Request permission

## Abstract:

Let $q$ be an odd number and $S_{q,0}(n)$ the difference between the number of $k<n$, $k\equiv 0\bmod q$, with an even binary digit sum and the corresponding number of $k<n$, $k\equiv 0\bmod q$, with an odd binary digit sum. A remarkable theorem of Newman says that $S_{3,0}(n)>0$ for all $n$. In this paper it is proved that the same assertion holds if $q$ is divisible by 3 or $q=4^N+1$. On the other hand, it is shown that the number of primes $q\le x$ with this property is $o(x/\log x)$. Finally, analoga for “higher parities” are provided.## References

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## Additional Information

**Michael Drmota**- Affiliation: Department of Geometry, Technical University of Vienna, Wiedner Hauptstrasse 8-10, A-1040 Vienna, Austria
- MR Author ID: 59890
- Email: michael.drmota@tuwien.ac.at
**Mariusz Skałba**- Affiliation: Institute of Mathematics, Warsaw University, Banacha 2, 02-097 Warsaw, Poland
- Email: skalba@mimuw.edu.pl
- Received by editor(s): July 6, 1995
- Received by editor(s) in revised form: December 2, 1997
- Published electronically: August 10, 1999
- Additional Notes: This work was supported by the Austrian Science Foundation, grant Nr. M 00233–MAT
- © Copyright 1999 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**352**(2000), 609-642 - MSC (1991): Primary 11B85; Secondary 11A63
- DOI: https://doi.org/10.1090/S0002-9947-99-02277-1
- MathSciNet review: 1491859