# Averaging almost periodic functions along exponential sequences

## TL;DR

This paper explores the averaging behavior of almost periodic functions along exponential sequences, highlighting their significance in spectral theory and uniform distribution, with results applicable to almost all real numbers.

## Contribution

It provides a self-contained account of averaging processes along exponential sequences, connecting almost periodic functions with spectral theory and uniform distribution.

## Key findings

- Averaging along exponential sequences converges for almost all real numbers.
- Results are metric and hold Lebesgue-almost everywhere.
- Applications include spectral analysis of inflation systems.

## Abstract

The goal of this expository article is a fairly self-contained account of some averaging processes of functions along sequences of the form $(\alpha^n x)^{}_{n\in\mathbb{N}}$, where $\alpha$ is a fixed real number with $| \alpha | > 1$ and $x\in\mathbb{R}$ is arbitrary. Such sequences appear in a multitude of situations including the spectral theory of inflation systems in aperiodic order. Due to the connection with uniform distribution theory, the results will mostly be metric in nature, which means that they hold for Lebesgue-almost every $x\in\mathbb{R}$.