On differences and sums of integers. I. (English) Zbl 0404.10029

Erdős, Paul; Sárkőzy, András

J. Number Theory 10, 430-450 (1978).

A set \(B=\{b_1,b_2,\dots,b_i\}\subset\{1,2,\dots,N\}\) is a difference intersector set if for any set \(A=\{a_1,a_2,\dots,a_j\}\subset\{1,2,\dots,N\}\), \(j=\varepsilon N\) 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 \(\left(\frac{a_x-a_y}p\right)=+1\), \(\left(\frac{a_r-a_s}p\right)=-1\), \(\left(\frac{a_t-a_u}p\right)=+1\), and \(\left(\frac{a_v-a_w}p\right)=-1\). The theorems are also used to establish that ”almost all” sequences form both difference and sum intersector sets.

Reviewer: M.M.Dodson

MSC:

11B83	Special sequences and polynomials
11B13	Additive bases, including sumsets
11P99	Additive number theory; partitions
11D85	Representation problems
11L03	Trigonometric and exponential sums, general

Keywords:

difference intersector set; sum intersector set; distribution quadratic residues; sequence of integers

References:

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