Quasirandom arithmetic permutations

In J.N. Cooper, Quasirandom permutations, 2002, to appear], the author introduced quasirandom permutations, permutations of 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 Erd?s-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.