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Title:
The Uniform Distribution of Sequences Generated by Iteration of Polynomials
Authors: 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
ePrint: https://eprint.iacr.org/2015/296
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