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Related Experiment Video

Updated: Jan 22, 2026

An Orbital Shaking Culture of Mammalian Cells in O-shaped Vessels to Produce Uniform Aggregates
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On Weyl products and uniform distribution modulo one.

Christoph Aistleitner1, Gerhard Larcher2, Friedrich Pillichshammer2

  • 11Institute for Analysis and Number Theory, Graz University of Technology, Graz, Austria.

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|July 2, 2019
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Summary

This study analyzes trigonometric products using evenly distributed points in the unit interval. We establish new bounds for these products, improving upon prior research and exploring special sequences.

Keywords:
Kronecker sequenceStar-discrepancyTrigonometric productvan der Corput sequence

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Area of Science:

  • Number Theory
  • Harmonic Analysis
  • Numerical Integration

Background:

  • Trigonometric products are fundamental in various mathematical fields.
  • Understanding their asymptotic behavior is crucial for analyzing sequences and distributions.
  • Previous work by Hlawka provided initial bounds for such products.

Purpose of the Study:

  • To investigate the asymptotic behavior of trigonometric products involving sequences of points.
  • To establish new, improved lower and upper bounds for these products.
  • To examine the behavior of these products for specific types of sequences, namely Kronecker and van der Corput sequences.

Main Methods:

  • Analysis of trigonometric products of the form $\prod_{k=1}^{N} 2\sin(\pi x_k)$ as $N \to \infty$.
  • Utilizing the concept of star-discrepancy to derive bounds for the trigonometric products.
  • Examining specific cases involving Kronecker and van der Corput sequences.

Main Results:

  • Derived matching lower and upper bounds for the trigonometric products.
  • These bounds are expressed in terms of the star-discrepancy of the underlying point sets.
  • The results improve upon the bounds previously established by Hlawka.

Conclusions:

  • The star-discrepancy effectively characterizes the asymptotic behavior of these trigonometric products.
  • The study provides a refined understanding of trigonometric products for uniformly distributed sequences.
  • Probabilistic analogues of the main results are also considered, suggesting broader applicability.