Paper 2015/296

The Uniform Distribution of Sequences Generated by Iteration of Polynomials

Emil Lerner

Abstract

Consider a collection $f$ of polynomials $f_i(x)$, $i=1, \ldots,s$, with integer coefficients such that polynomials $f_i(x)-f_i(0)$, $i=1, \ldots,s$, are linearly independent. Denote by $D_m$ the discrepancy for the set of points $\left(\frac{f_1(x) \bmod m}{m},\ldots,\frac{f_s(x) \bmod m}{p^n}\right)$ for all $x \in \{0,1,\ldots,m\}$, where $m=p^n$, $n \in N$, and $p$ is a prime number. We prove that $D_m\to 0$ as $n\to\infty$, and $D_m<c_1 (\log \log m)^{-c_2}$, where $c_1$ and $c_2$ are positive constants that depend only on the collection of $f_i$. As a corollary, we obtain an analogous result for iterations of any polynomial (with integer coefficients) whose degree exceeds~1. Certain results on the uniform distribution were known earlier only for some classes of polynomials with $s \leqslant 3$