Exponential sums with automatic sequences

Sary Drappeau (1) , Clemens Müllner (2)

1 I2M - Institut de Mathématiques de Marseille
2 Institut für Diskrete Mathematik und Geometrie [Wien]

We show that automatic sequences are asymptotically orthogonal to periodic exponentials of type $e_q(f(n))$, where $f$ is a rational fraction, in the P\'olya-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.

Domaines: Théorie des nombres [math.NT]

Sary Drappeau, Clemens Müllner. Exponential sums with automatic sequences. Acta Arithmetica, 2018, 185, pp.81-99. ⟨10.4064/aa171002-20-3⟩. ⟨hal-01847933⟩