Discrepancy estimates for index-transformed uniformly distributed sequences

TL;DR

This paper establishes discrepancy bounds for index-transformed uniformly distributed sequences, including van der Corput, Halton, and $(t,s)$-sequences, with applications to sequences indexed by functions like $loor{n^{eta}}$, providing insights into their uniformity properties.

Contribution

It introduces new discrepancy bounds for index-transformed sequences, especially those indexed by sum-of-digits and power functions, advancing understanding of their distribution properties.

Findings

Derived tight bounds for discrepancy of index-transformed sequences

Analyzed discrepancy for sequences indexed by functions like $loor{n^{eta}}$

Provided bounds applicable to van der Corput, Halton, and $(t,s)$-sequences

Abstract

In this paper we show discrepancy bounds for index-transformed uniformly distributed sequences. From a general result we deduce very tight lower and upper bounds on the discrepancy of index-transformed van der Corput-, Halton-, and $(t,s)$-sequences indexed by the sum-of-digits function. We also analyze the discrepancy of sequences indexed by other functions, such as, e.g., $\lfloor n^{\alpha}\rfloor$ with $0 < \alpha < 1$.