On differences and sums of integers, I

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Abstract

A set {b1,b2,...,bi} ⊂ {1,2,...,N} is said to be a difference intersector set if {a1,a2,...,as} ⊂ {1,2,...,N}, j > ε{lunate}N imply the solvability of the equation ax - ay = b′; the notion of sum intersector set is defined similarly. The authors prove two general theorems saying that if a set {b1,b2,...,bi} is well distributed simultaneously among and within all residue classes of small moduli then it must be both difference and sum intersector set. They apply these theorems to investigate the solvability of the equations (ax - ay p = + 1, (au - av p) = - 1, (ar + as p) = + 1, (at + az p) = - 1 (where ( a p) denotes the Legendre symbol) and to show that "almost all" sets form both difference and sum intersector sets.

Original languageEnglish
Pages (from-to)430-450
Number of pages21
JournalJournal of Number Theory
Volume10
Issue number4
DOIs
Publication statusPublished - Nov 1978

ASJC Scopus subject areas

  • Algebra and Number Theory

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