Quasirandom permutations

Chung and Graham (J. Combin. Theory Ser. A 61 (1992) 64) define quasirandom subsets of Zn to be those with any one of a large collection of equivalent random-like properties. We weaken their definition and call a subset of Zn ε-balanced if its discrepancy on each interval is bounded by εn. A quasirandom permutation, then, is one which maps each interval to a highly balanced set. In the spirit of previous studies of quasirandomness, we exhibit several random-like properties which are equivalent to this one, including the property of containing (approximately) the expected number of subsequences of each order-type. We present a construction for a family of strongly quasirandom permutations, and prove that this construction is essentially optimal, using a result of Schmidt on the discrepancy of sequences of real numbers.