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Bounded error uniformity of the linear flow on the torus

  • Bence Borda
Article
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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.

Keywords

Continuous uniform distribution Set of bounded remainder Discrepancy 

Mathematics Subject Classification

11K38 11J87 

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Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Alfréd Rényi Institute of MathematicsHungarian Academy of SciencesBudapestHungary

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