Metric uniform distribution on analytic curves

Abstract

We obtain multidimensional metric uniform distribution results involving sequences in Rk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^k$$\end{document} 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 a1,⋯,ak:N→Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_1,\dots ,a_k:\mathbb {N}\rightarrow \mathbb {Z}$$\end{document} the set {(x1,⋯,xk)∈Rk:(a1(n)x1,⋯,ak(n)xk)n∈Nis uniformly distributed inTk}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Big \{(x_1,\dots ,x_k)\in \mathbb {R}^k:\big (a_1(n)x_1,\dots ,a_k(n)x_k\big )_{n\in \mathbb {N}}\text { is uniformly distributed in }\mathbb {T}^k\Big \} \end{aligned}$$\end{document}has full Lebesgue measure inside any non-degenerate analytic curve γ⊂Rk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \subset \mathbb {R}^k$$\end{document}.