Non-Abelian homomorphism testing, and distributions close to their self-convolutions

In this paper, we study two questions related to the problem of testing whether a function is close to a homomorphism. For two finite groups G, H (not necessarily Abelian), an arbitrary map f : G → H, and a parameter 0 < ε < 1, say that f is ε-close to a homomorphism if there is some homomorphism g such that g and f differ on at most ε|G| elements of G, and say that f is ε-far otherwise. For a given f and ε, a homomorphism tester should distinguish whether f is a homomorphism, or if f is ε-far from a homomorphism. When G is Abelian, it was known that the test which picks O(1/ε) random pairs x,y and tests that f(x) + f(y) = f(x + y) gives a homomorphism tester. Our first result shows that such a test works for all groups G. Next, we consider functions that are close to their self-convolutions. Let A = {a_g|g ε G} be a distribution on G. The self-convolution of A, A′ = {a′_g\g ε G}, is defined by a′_x = ∑_y,zεG;yz=xayaz. It is known that A = A′ exactly when A is the uniform distribution over a subgroup of G. We show that there is a sense in which this characterization is robust - that is, if A is close in statistical distance to A′, then A must be close to uniform over some subgroup of G.