On uniform distribution of $\alpha\beta$-orbits

TL;DR

This paper proves that for almost all choices of irrational parameters, the combined orbit of a random walk is uniformly distributed modulo one, with exponential sums exhibiting square root cancellation, and identifies a large exceptional set lacking this property.

Contribution

It establishes almost sure uniform distribution of $eta$-orbits under random walks and analyzes the size of the exceptional set where this fails.

Findings

Almost sure uniform distribution of $eta$-orbits modulo one.

Exponential sums along the orbit show square root cancellation.

The exceptional set is large in topological and Hausdorff dimension terms.

Abstract

Let $\alpha, \beta \in (0,1)$ such that at least one of them is irrational. We take a random walk on the real line such that the choice of $\alpha$ and $\beta$ has equal probability $1/2$. We prove that almost surely the $\alpha\beta$-orbit is uniformly distributed module one, and the exponential sums along its orbit has the square root cancellation. We also show that the exceptional set in the probability space, which does not have the property of uniform distribution modulo one, is large in the terms of topology and Hausdorff dimension.