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**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 = **{**b_{1},b_{2},...,b_{i}**}**\subset**{**1,2,...,N**}** is a difference intersector set if for any set A = **{**a_{1},a_{2},...,a_{j}**}**\subset**{**1,2,...,N**}**, j = \epsilon N the equation a_{x}-a_{y} = 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{a_{x}-a_{y}}p**)** = +1, **(**\frac{a_{r}-a_{s}}p**)** = -1, **(**\frac{a_{t}-a_{u}}p**)** = +1, and **(**\frac{a_{v}-a_{w}}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

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag