Metric uniform distribution on analytic curves

Abstract

We obtain multidimensional metric uniform distribution results involving sequences in ${\mathbb{R}}^k$ parametrized by analytic curves. Our theorems extend the classical theorems of Weyl and Koksma in a variety of ways. One of our main results implies that for any injective sequences $a_1,\dots ,a_k:\mathbb{N}\to \mathbb{Z}$ the set $$\{(x_1,\dots ,x_k)\in {\mathbb{R}}^k:(a_1(n)x_1,\dots ,a_k(n)x_k{)}_{n\in \mathbb{N}}\text{ is uniformly distributed in }{\mathbb{T}}^k\}$$ has full Lebesgue measure inside any non-degenerate analytic curve $\gamma \subset {\mathbb{R}}^k$.

Appendix by Jim Wright. 36 pages. V2 incorporates comments of referee