Approximation by uniformly distributed sequences

TL;DR

This paper investigates how well real points can be approximated by uniformly distributed sequences, establishing a dichotomy law, connecting uniform distribution with ubiquity, and deriving Hausdorff dimension results using advanced tools.

Contribution

It introduces a new connection between uniform distribution and ubiquity, and applies weighted mass transference principles to approximation by uniformly distributed sequences.

Findings

Proves a Khintchine-type 0-1 law for approximation by uniformly distributed sequences.

Shows that bounds on discrepancy imply the ubiquity property.

Derives Hausdorff dimension results for weighted approximation sets.

Abstract

We consider approximation properties of real points by uniformly distributed sequences. Under some assumptions on the approximation functions, we prove a Khintchine-type $0$-$1$ dichotomy law. We establish a new connection between uniform distribution and the ubiquity property. Namely, we show that a bound on the discrepancy of the sequence implies the ubiquity property, which helps to obtain divergence results. We further obtain Hausdorff dimension results for weighted sets. The key tools in proving these results are the weighted ubiquitous systems and weighted mass transference principle introduced recently by Kleinbock \& Wang, and Wang \& Wu respectively.