On the Dynamical System Generated by the M\"obius Transformation at Smooth Times

TL;DR

This paper investigates the distribution of sequences generated by a Möbius transformation over finite fields at smooth times, providing estimates of exponential sums and insights into their dynamical properties.

Contribution

It introduces new estimates for exponential sums of Möbius transformation sequences at smooth times over finite fields, advancing understanding of their distribution.

Findings

Derived nontrivial bounds for exponential sums

Analyzed distribution properties at smooth times

Enhanced understanding of Möbius dynamical systems

Abstract

We study the distribution of the sequence of the first $N$ elements of the discrete dynamical system generated by the M\"obius transformation $x \mapsto (\alpha x + \beta)/(\gamma x + \delta)$ over a finite field of $p$ elements at the moments of time that correspond to $Q$-smooth numbers, that is, to numbers composed out of primes up to $Q$. In particular, we obtain nontrivial estimates of exponential sums with such sequences.