On the sequence $\alpha n!$

TL;DR

This paper proves the existence of a real number alpha for which the discrepancy of the sequence {alpha n!} remains bounded by a logarithmic function, demonstrating a specific uniform distribution property.

Contribution

It establishes the existence of a real number alpha such that the discrepancy of the sequence {alpha n!} is logarithmically bounded for all N, a novel result in uniform distribution theory.

Findings

Discrepancy D_N of the sequence {alpha n!} is O(log N) for some alpha.

Existence of such alpha demonstrates a new uniform distribution property.

Sequence {alpha n!} exhibits controlled irregularity in distribution.

Abstract

We prove that there exists $\alpha \in \mathbb{R}$ such that for any $N$ the dicrepancy $D_N$ of the sequence $\{\alpha n !\} ,\, 1\le n \le N$ satisfies $ D_N = O(\log N)$.