The Uniform Distribution of Sequences Generated by Iterated Polynomials
- Авторы: Lerner E.1
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Учреждения:
- Moscow State University, Faculty of Computational Mathematics and Cybernetics
- Выпуск: Том 11, № 4 (2019)
- Страницы: 280-298
- Раздел: Research Articles
- URL: https://journals.rcsi.science/2070-0466/article/view/201280
- DOI: https://doi.org/10.1134/S2070046619040034
- ID: 201280
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Аннотация
In the paper we show that given a polynomial f over ℤ = 0, ±1, ±2, ..., deg f ⩾ 2, the sequence x, f(x), f(f(x)) = f(2)(x), ..., where x is m-adic integer, produces a uniformly distributed set of points in every real unit hypercube under a natural map of the space ℤm of m-adic integers onto unit real interval. Namely, let m, s ∈ ℕ = {1, 2, 3, ...}, m > 1, let κn have a discrete uniform distribution on the set {0, 1, ..., mn - 1. We prove that with n tending to infinity random vectors
\(\left(\frac{\kappa_n}{m^n}, \frac{f(\kappa_n){\rm{mod}} m^n}{m^n}, \ldots, \frac{f^{(s-1)}(\kappa_n) {\rm{mod}} m^n}{m^n}\right)\)
weakly converge to a vector having a continuous uniform distribution in the s-dimensional unit hypercube. Analogous results were known before only for the case when s ⩽ 3 and f is a quadratic polynomial (deg f = 2).Об авторах
Emil Lerner
Moscow State University, Faculty of Computational Mathematics and Cybernetics
Автор, ответственный за переписку.
Email: neex.emil@gmail.com
Россия, Moscow