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).

中文翻译：

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由迭代多项式生成的序列的均匀分布

在论文中，我们证明给定多项式 f 超过 ℤ = 0, ±1, ±2, ..., deg f ⩾ 2, 序列 x, f(x), f(f(x)) = f(2 )(x), ...，其中 x 是 m-adic 整数，在 m-adic 整数空间 ℤm 到单位实数区间的自然映射下，在每个实单位超立方体中产生均匀分布的点集。即，令 m, s ∈ ℕ = {1, 2, 3, ...}, m > 1，令 κn 在集合 {0, 1, ..., mn - 1 上具有离散均匀分布。我们证明当 n 趋向于无穷大随机向量 $$\left(\frac{\kappa_n}{m^n}, \frac{f(\kappa_n){\rm{mod}} m^n}{m^n}, \ldots, \frac{f^{(s-1)}(\kappa_n) {\rm{mod}} m^n}{m^n}\right)$$ 弱收敛到具有连续均匀分布的向量在 s 维单位超立方体中。类似的结果以前只在 s ⩽ 3 和 f 是二次多项式 (deg f = 2) 的情况下才知道。