On Differences and Sums of Integers , I

imply the solvability of the equation a s-av = b' ; the notion of sum intersector set is defined similarly. The authors prove two general theorems saying that if a set {b, , b 2 BLOCKINbi} 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 (a .-ay/p = + 1, (a,,.-a " lp) _-1, (a, + a,lp) _ + 1, (a t + a.1p) _-1 (where (a/p) denotes the Legendre symbol) and to show that "almost all" sets form both difference and sum intersector sets. 1 Throughout this paper, we use the following notations : C l , C2 ,. .. will denote positive absolute constants. We write ex = exp(x). For real a, we put e(a) = e 2 ~i. . If p is a prime number and n is an integer then we denote the least nonnegative residue of n modulo p by r aEA beB acn b<n If the infinite sequence B = {b, , b 2 ,. . .} is such that the equation-a y = b, (1) is solvable for every infinite sequence A = {a, , a2 ,. . .} of positive lower (asymptotic) density (i .e ., B intersects the difference set of each of these 430