Metric uniform distribution on analytic curves
Abstract
Abstract We obtain multidimensional metric uniform distribution results involving sequences in $$\mathbb {R}^k$$ 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 $$a_1,\dots ,a_k:\mathbb {N}\rightarrow \mathbb {Z}$$ a 1 , ⋯ , a k : N → Z the set $$\begin{aligned} \Big \{(x_1,\dots ,x_k)\in \mathbb {R}^k:\big (a_1(n)x_1,\dots ,a_k(n)x_k\big )_{n\in \mathbb {N}}\text { is uniformly distributed in }\mathbb {T}^k\Big \} \end{aligned}$$ { ( 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 $$\gamma \subset \mathbb {R}^k$$ γ ⊂ R k .
Fields of Study & Subjects
Functional Equations Stability Results
advanced mathematical theories
Publication details
- Journal / source
- Research in the Mathematical Sciences
- Publisher
- Springer Science and Business Media LLC
- Volume
- 13
- Issue
- 3
- Pages
- 77–77
- Publication date
- 1 September 2026
- Language
- EN