Bounded error uniformity of the linear flow on the torus

TL;DR

This paper proves bounded error uniformity of certain linear flows on the torus with algebraic directions, highlighting fundamental differences between continuous and discrete uniform distribution theories.

Contribution

It combines Fourier analysis and Schmidt's subspace theorem to establish bounded error uniformity for algebraic directions, extending uniform distribution results.

Findings

Bounded error uniformity holds for linear flows with algebraic directions.

No van Aardenne–Ehrenfest type theorem exists for the discrepancy of continuous curves.

Demonstrates a fundamental difference between continuous and discrete uniform distribution.

Abstract

A linear flow on the torus $\mathbb{R}^d / \mathbb{Z}^d$ is uniformly distributed in the Weyl sense if the direction of the flow has linearly independent coordinates over $\mathbb{Q}$. In this paper we combine Fourier analysis and the subspace theorem of Schmidt to prove bounded error uniformity of linear flows with respect to certain polytopes if, in addition, the coordinates of the direction are all algebraic. In particular, we show that there is no van Aardenne--Ehrenfest type theorem for the mod $1$ discrepancy of continuous curves in any dimension, demonstrating a fundamental difference between continuous and discrete uniform distribution theory.