The coin problem for product tests

Let Xm,ε be the distribution over m bits (X1, . . . , Xm) where the Xi are independent and each Xi equals 1 with probability (1 + ε)/2 and 0 with probability (1 − ε)/2. We consider the smallest value ε∗ of ε such that the distributions Xm,ε and Xm,0 can be distinguished with constant advantage by a function f : {0, 1}m → S which is the product of k functions f1, f2, . . . , fk on disjoint inputs of n bits, where each fi : {0, 1}n → S and m = nk. We prove that ε∗ = Θ(1/ √ n log k) if S = {0, 1} and if S = {−1, 1}, while ε∗ = Θ(1/ √ nk) if S is the set of unit-norm complex numbers.