Quasirandom arithmetic permutations Original Research Article

In [J.N. Cooper, Quasirandom permutations, 2002, to appear], the author introduced quasirandom permutations, permutations of image 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 Erdős–Turán inequality, as well as by other means. We apply our results on Sós permutations to make progress on a number of questions relating to the sequence of fractional parts of multiples of an irrational. Several intriguing open problems are presented throughout the discussion.