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Quasirandom Permutations

Joshua N. Cooper
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Abstract

Chung and Graham define quasirandom subsets of $\mathbb{Z}_n$ to be those with any one of a large collection of equivalent random-like properties. We weaken their definition and call a subset of $\mathbb{Z}_n$ $ε$-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 provide a few applications of these results, present a construction for a family of strongly quasirandom permutations, and prove that this construction is essentially optimal, using a result of W. Schmidt on the discrepancy of sequences of real numbers.

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Joshua N. Cooper. 2002-10-31. Quasirandom Permutations. https://arxiv.org/abs/math/0211001

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