The Uniform Distribution of Sequences Generated by Iterated Polynomials

Abstract

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 ℤ_m of m-adic integers onto unit real interval. Namely, let m, s ∈ ℕ = {1, 2, 3, ...}, m > 1, let κ_n 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).