An improved bound on correlation between polynomials over Z_m and MOD_q

Let m, q > 1 be two integers that are co-prime and A be any subset of Zm. Let P be any multi-variate polynomial of degree d in n variables over Zm. We show that the MODq boolean function on n variables has correlation at most exp(−Ω(n/(m2 ))) with the boolean function f defined by f(x) = 1 iff P (x) ∈ A for all x ∈ {0, 1}. This improves on the bound of exp(−Ω(n/(m2))) obtained in the breakthrough work of Bourgain [3] and Green et al. [9]. Our calculation is also slightly shorter than theirs. Our result immediately implies the bound of exp(−Ω(n/4)) for the special case of m = 2. This bound was first reported in the recent work of Viola [11]. [11] states that it is not clear how to extend their method to general m.