  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.
