Quasirandom Arithmetic Permutations

Abstract

Previously, the author introduced quasirandom permutations, permutations of $\mathbb{Z}_n$ which map intervals to sets with low discrepancy. Here we show that several natural number-theoretic permutations are quasirandom, some very strongly so. Quasirandomness is established via discrete Fourier analysis and the Erdos-Turan inequality, as well as by other means. We apply our results on Sos permutations to make progress on a number of questions relating to the sequence of fractional parts of multiples of an irrational. Several intriguing new open problems are presented throughout the discussion.