A note on Weyl's equidistribution theorem.

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 $p\in (1,\infty )$ the ${\ell }^p$ norms of integer vectors are equidistributed mod 1.