# The discrepancy of $(n_kx)$ with respect to certain probability measures

## TL;DR

This paper proves a conjecture regarding the discrepancy of lacunary sequences with respect to certain probability measures, showing almost sure bounds on the normalized discrepancy and improving previous results on related Diophantine approximation products.

## Contribution

It establishes almost sure bounds on the discrepancy of lacunary sequences under specific measures, confirming a conjecture and refining earlier results in Diophantine approximation.

## Key findings

- Almost sure bounds on the discrepancy for lacunary sequences
- Validation of a conjecture by Haynes, Jensen, and Kristensen
- Improved bounds on products involving Diophantine approximation

## Abstract

Let $(n_k)_{k=1}^{\infty}$ be a lacunary sequence of integers. We show that if $\mu$ is a probability measure on $[0,1)$ such that $|\widehat{\mu}(t)|\leq c|t|^{-\eta}$, then for $\mu$-almost all $x$, the discrepancy $D_N(n_kx)$ satisfies \begin{equation*} \frac{1}{4} \leq \limsup_{N\to\infty}\frac{N D_N(n_kx)}{\sqrt{N\log\log N}} \leq C \end{equation*} for some constant $C>0$, proving a conjecture of Haynes, Jensen and Kristensen. This allows a slight improvement on their previous result on products of the form $q\|q\alpha\| \|q\beta-\gamma\| $.