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