The uniform distribution of sequences generated by iterated polynomials

TL;DR

This paper proves that sequences generated by iterated polynomials modulo m^n become uniformly distributed in the s-dimensional unit cube as n approaches infinity, extending previous results to more general polynomials.

Contribution

It generalizes the uniform distribution result for sequences from polynomial iterates to broader classes of polynomials with degree greater than one.

Findings

Sequences converge to uniform distribution in the unit cube

Extends previous results to higher degrees and dimensions

Provides theoretical foundation for polynomial pseudorandom number generators

Abstract

Assume that $m,s\in\mathbb N$, $m>1$, while $f$ is a polynomial with integer coefficients, $\text{deg}~f>1$, $f^{(i)}$ is the $i$th iteration of the polynomial $f$, $\kappa_n$ has a discrete uniform distribution on the set $\{0,1,\ldots,m^n - 1\}$. We are going to prove that with $n$ tending to infinity random vectors $\left(\frac{\kappa_n}{m^n},\frac{f(\kappa_n) \bmod m^n}{m^n},\ldots,\frac{f^{(s - 1)}(\kappa_n) \bmod m^n}{m^n}\right)$ weakly converge to a vector having a continuous uniform distribution in the $s$-dimensional unit cube. Analogous results were obtained earlier only for some classes of polynomials with $s\leqslant 3, \text{deg}~f = 2$. The mentioned vectors represent sequential pseudorandom numbers produced by a polynomial congruential generator modulo $m^n$.