The Number of Solutions of Congruences in Incomplete Residue Systems

Abstract

Let denote the set of points x = (x1, . . . , xn) with integral co-ordinates in Euclidean n-space. For any fixed integer m ≥ 2, let C = C(m) be the set of such points in the cube 0 ≤ xi < m (i = 1, 2, . . . , n) and let be any subset of C. Suppose that f(x) is any single-valued, integral-valued function, defined for all x ∊ . We consider solutions x ∊ of the congruence