Variants of the Littlewood conjecture, their connection to uniformly distributed sequences, and the exact order of the discrepancy of van der Corput--Kronecker-type sequences

TL;DR

This paper explores variants of the Littlewood conjecture in relation to uniformly distributed sequences and determines the precise order of discrepancy for certain sequences, supporting longstanding conjectures in uniform distribution theory.

Contribution

It establishes the exact order of discrepancy for van der Corput--Kronecker-type sequences based on recent counterexamples, advancing understanding of uniform distribution.

Findings

Confirmed the conjectured upper bound for discrepancy in these sequences.

Provided new insights into the connection between Littlewood conjecture variants and uniform distribution.

Supported the optimality of the rac{ ext{log}^s N}{N} discrepancy bound.

Abstract

The aims of this paper are twofold. First, it discusses the Littlewood conjecture and its variants with respect to uniformly distributed sequences. The second aim is to determine the exact order of the discrepancy of the van der Corput--Kronecker-type sequences which are based on recent counterexamples to the $X$-adic Littlewood conjecture over a finite field. Our result on the exact order of the discrepancy supports the well-established conjecture in the theory of uniform distribution, which states that $D_N\leq c \frac{\log^s N}{N}$, with $c>0$ for all $N>1$ is the best possible upper bound for the discrepancy $D_N$ of a sequence in $[0,1)^s$.