Metric uniform distribution on analytic curves: With an Appendix by Jim Wright.

We obtain multidimensional metric uniform distribution results involving sequences in 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 , ⋯ , a k : N → Z the set { ( x 1 , ⋯ , x k ) ∈ R k : ( a 1 ( n ) x 1 , ⋯ , a k ( n ) x k ) n ∈ N is uniformly distributed in T k } has full Lebesgue measure inside any non-degenerate analytic curve γ ⊂ R k .