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 = [−1,1], while ϵ* = Θ(1/√nk) if S is the set of unit-norm complex numbers.