Mathematics – Combinatorics
Scientific paper
2002-10-31
Mathematics
Combinatorics
30 pages, 2 figures, submitted to JCTA
Scientific paper
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 $\mathbb{Z}_n$ $\epsilon$-balanced if its discrepancy on each interval is bounded by $\epsilon n$. 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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