A note on Weyl's equidistribution theorem

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.