On the distribution of multiplicative translates of sets of residues (mod p)

J. Håstad, J. C. Lagarias, A. M. Odlyzko

Research output: Contribution to journalArticlepeer-review

Abstract

Let R be a set of r distinct nonzero residues modulo a prime p, and suppose that the random variable a is drawn with the uniform distribution from {1, 2,…, p - 1}. We show for all sets R that (p - 2)/(2r) ≤ E[min[aR]] ≤ 100 p/r1/2, where in the set aR each integer is identified with its least positive residue modulo p. We give examples where E[min[aR]] ≤ 0.8 p/r and E[min[aR]] ≥ 0.4 p(log r)/r. We conjecture that E[min[aR]] ≪ p/r1 - ε{lunate} holds for a wide range of r. These results are applicable to the analysis of certain randomization procedures.

Original languageEnglish (US)
Pages (from-to)108-122
Number of pages15
JournalJournal of Number Theory
Volume46
Issue number1
DOIs
StatePublished - Jan 1994

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