ABSTRACT
Let N be an integer ≥ 1, and, for each prime p, let νp be a real number with 0 < νp ≤ 1. Let A be a subset of Λ = Zn, such that for all primes p,
|Ap| ≤ νppn,
where Ap ⊂ Λ/pΛ denotes the reduction of A mod p. Given a vector x = (x1, . . . , xn) ∈ Rn, and N ∈ R, we denote by A(x,N) the set of points in A which are contained in the cube of side length N centered at x, i.e.,
A(x,N) = {(a1, . . . , an) ∈ A | |xi − ai| ≤ N/2}.
