Exponential sums with automatic sequences
Volume 185 / 2018
Acta Arithmetica 185 (2018), 81-99
MSC: Primary 11L07, 11B85; Secondary 11L05, 11L26.
DOI: 10.4064/aa171002-20-3
Published online: 15 June 2018
Abstract
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.
