How close can we come to a parity function when there isn't one?

TL;DR

This paper investigates how closely functions on a group can resemble homomorphisms when no true homomorphism exists, providing bounds based on group representation theory and specific examples like the alternating group.

Contribution

It establishes probabilistic bounds on near-homomorphisms for groups without actual homomorphisms, linking these bounds to the group's irreducible representations and subgroup structures.

Findings

Probability bound for approximate homomorphisms based on irreducible representation dimension

Explicit construction of functions on A_n with near-homomorphic properties

Bounds on how close such functions can get to true homomorphisms, between O(n^{-1/2}) and Omega(n^{-2})

Abstract

Consider a group G such that there is no homomorphism f:G to {+1,-1}. In that case, how close can we come to such a homomorphism? We show that if f has zero expectation, then the probability that f(xy) = f(x) f(y), where x, y are chosen uniformly and independently from G, is at most 1/2(1+1/sqrt{d}), where d is the dimension of G's smallest nontrivial irreducible representation. For the alternating group A_n, for instance, d=n-1. On the other hand, A_n contains a subgroup isomorphic to S_{n-2}, whose parity function we can extend to obtain an f for which this probability is 1/2(1+1/{n \choose 2}). Thus the extent to which f can be "more homomorphic" than a random function from A_n to {+1,-1} lies between O(n^{-1/2}) and Omega(n^{-2}).