Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 404.10029
Autor: Erdös, Paul; Sárközy, András
Title: On differences and sums of integers. I. (In English)
Source: J. Number Theory 10, 430-450 (1978).
Review: A set B = {b1,b2,...,bi}\subset{1,2,...,N} is a difference intersector set if for any set A = {a1,a2,...,aj}\subset{1,2,...,N}, j = \epsilon N the equation ax-ay = b has a solution. The notion of a sum intersector set is defined similary. Using exponential sum techniques, the authors prove two theorems which in essence imply that a set which is well-distributed within and amongst all residue classes of small modules is both a difference and a sum intersector set. The regularity of the distribution of the non-zero quadratic residues (mod p) allows the theorems to be used to investigate the solubility of the equations (\frac{ax-ay}p) = +1, (\frac{ar-as}p) = -1, (\frac{at-au}p) = +1, and (\frac{av-aw}p) = -1. The theorems are also used to establish that ''almost all'' sequences form both difference and sum intersector sets.
Reviewer: M.M.Dodson
Classif.: * 11B83 Special sequences of integers and polynomials
11B13 Additive bases
11P99 Additive number theory
11D85 Representation problems of integers
11L03 Trigonometric and exponential sums, general
Keywords: difference intersector set; sum intersector set; distribution quadratic residues; sequence of integers
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