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Exponential sums with automatic sequences

Volume 185 / 2018

Sary Drappeau, Clemens Müllner Acta Arithmetica 185 (2018), 81-99 MSC: Primary 11L07, 11B85; Secondary 11L05, 11L26. DOI: 10.4064/aa171002-20-3 Published online: 15 June 2018


We show that automatic sequences are asymptotically orthogonal to periodic exponentials of type~${\rm e}_q(f(n))$, where $f$ is a rational fraction, in the Pólya–Vinogradov range. This applies to Kloosterman sums, and may be used to study solubility of congruence equations over automatic sequences. We obtain this as consequence of a general result, stating that sums over automatic sequences can be bounded effectively in terms of two-point correlation sums over intervals.


  • Sary DrappeauAix Marseille Université
    CNRS, Centrale Marseille
    I2M UMR 7373
    13453 Marseille, France
  • Clemens MüllnerInstitut für Diskrete Mathematik
    und Geometrie
    TU Wien
    Wiedner Hauptstr. 8–10
    1040 Wien, Austria

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