AND testing and robust judgement aggregation

A function f : 0, 1 n → 0, 1 is called an approximate AND-homomorphism if choosing x, y ∈ 0, 1 n randomly, we have that f (x ∧ y) = f (x) ∧ f (y) with probability at least 1ε, where x ∧ y = (x 1 ∧ y 1 , . . . , x n ∧ y n ). We prove that if f : 0, 1 n → 0, 1 is an approximate AND-homomorphism, then f is δ-close to either a constant function or an AND function, where δ(ε) → 0 as ε → 0. This improves on a result of Nehama, who proved a similar statement in which δ depends on n. Our theorem implies a strong result on judgement aggregation in computational social choice. In the language of social choice, our result shows that if f is ε-close to satisfying judgement aggregation, then it is δ(ε)-close to an oligarchy (the name for the AND function in social choice theory). This improves on Nehama's result, in which δ decays polynomially with n. Our result follows from a more general one, in which we characterize approximate solutions to the eigenvalue equation Tf = λg, where T is the downwards noise operator Tf (x) = E y [f (x ∧ y)], f is [0, 1]-valued, and g is 0, 1-valued. We identify all exact solutions to this equation, and show that any approximate solution in which Tf and λg are close is close to an exact solution.