A Note about Weyl Equidistribution Theorem

Abstract

H. Weyl proved in \cite{Weyl} that integer evaluations of polynomials are equidistributed mod 1 whenever at least one of the non-free coefficients 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 strengths the main result of Arhipov-Karacuba-\v{C}ubarikov in \cite{PolWeyl}. We prove this analogue as a Corollary of a Theorem that guarantees equidistribution of lattice evaluations mod 1 for all functions which satisfy some restrains on their derivatives. Another Corollary we prove is that for $p\in(1,\infty)$ the $\ell^p$ norms of integer vectors are equidistributed mod 1.