# A note on Weyl’s equidistribution theorem

## TL;DR

This paper extends Weyl's equidistribution theorem to higher dimensions and proves new equidistribution results for polynomial and norm evaluations.

## Contribution

A higher-dimensional generalization of Weyl’s equidistribution theorem and a new result on the equidistribution of ℓ^p norms of integer vectors mod 1.

## Key findings

- Polynomial evaluations on lattice points are equidistributed mod 1 if at least one non-free coefficient is irrational.
- ℓ^p norms of integer vectors are equidistributed mod 1 for p in (1, ∞).
- The result improves upon a previous theorem by Arhipov et al.

## Abstract

H. Weyl proved in Weyl (Eins Math Ann 77(3):313–352, 1916) that integer evaluations of polynomials are equidistributed mod 1 whenever at least one of the non-free coefficients (namely a coefficient of a monomial of degree at least 1) is irrational. We use Weyl’s result to prove a higher dimensional analogue of this fact. Namely, we prove that evaluations of polynomials on lattice points are equidistributed mod 1 whenever at least one of the non-free coefficients is irrational. This result improves the main result of Arhipov et al. (Mat Zametki 25(1):3–14, 157, 1979). We prove this analogue as a corollary of a theorem that guarantees equidistribution of grid evaluations mod 1 for all functions which satisfy some restraints on their derivatives. Another corollary we prove is that for \documentclass[12pt]{minimal}
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\begin{document}$$p\in (1,\infty )$$\end{document}p∈(1,∞) the \documentclass[12pt]{minimal}
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\begin{document}$$\ell ^p$$\end{document}ℓp norms of integer vectors are equidistributed mod 1.