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 $$\Big\{(x_1,\dots,x_k)\in{\mathbb R}^k:\big(a_1(n)x_1,\dots,a_k(n)x_k\big)_{n\in{\mathbb N}}\text{ is uniformly distributed in }{\mathbb T}^k\Big\}$$ has full Lebesgue measure inside any non-degenerate analytic curve $\gamma\subset{\mathbb R}^k$.