Congruence properties of the Ω-function on sumsets

Throughout this paper, we shall use the following notation: c1,c2,. . . denote positive absolute constants. Z, N and N0 denote the set of integers, positive integers and non-negative integers respectively. The cardinality of a set S is denoted by |S|. bxc and {x} denote the integer part and the fractional part of x and ‖x‖ denotes the distance from x to the nearest integer: ‖x‖ = min({x}, 1 − {x}). We write e = e(α). If f(n) = O(g(n)), then we write f(n) g(n); if the implied constant depends on a certain parameter c, then we write f(n) c g(n). A,B, . . . denote subsets of N0 and A+ B denotes the set of the non-negative integers n that can be represented in the form n = a + b with a ∈ A, b ∈ B. ω(n) denotes the number of distinct prime factors of n and Ω(n) denotes the number of prime factors of n counted with multiplicity. λ(n) is the Liouville function: λ(n) = (−1). The divisor function is denoted by τ(n).