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A note on Weyl's equidistribution theorem.

Yuval Yifrach1

  • 1University of Zurich, Zurich, Switzerland.

Monatshefte Fur Mathematik
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PubMed
Summary
This summary is machine-generated.

Polynomials with irrational coefficients exhibit equidistribution modulo 1 for lattice point evaluations. This extends Weyl

Keywords:
Distribution modulo oneEquidistributionHaar measureHomogeneous functionsLattice pointsMultivariable polynomialsWeak convergenceWeyl’s theorem

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Area of Science:

  • Number Theory
  • Diophantine Approximation
  • Harmonic Analysis

Background:

  • H. Weyl's theorem on equidistribution of polynomial values modulo 1 for irrational coefficients.
  • Prior work by Arhipov et al. on higher dimensional analogues.

Purpose of the Study:

  • To prove a higher dimensional analogue of Weyl's equidistribution theorem.
  • To establish equidistribution of polynomial evaluations on lattice points when at least one non-free coefficient is irrational.

Main Methods:

  • Leveraging Weyl's original result.
  • Proving a general theorem on equidistribution of grid evaluations for functions with specific derivative properties.
  • Applying this theorem as a corollary.

Main Results:

  • Demonstrated that polynomial evaluations on lattice points are equidistributed modulo 1 if any non-free coefficient is irrational.
  • Improved upon the main result of Arhipov et al.
  • Showcased equidistribution of L^p norms of integer vectors modulo 1.

Conclusions:

  • The study generalizes Weyl's equidistribution theorem to higher dimensions for polynomial lattice point evaluations.
  • The findings have implications for number theory and diophantine approximation.
  • New results on the distribution of vector norms are also presented.