Proving the conjecture of O’Donnell in certain cases and disproving its general validity

Abstract

For a function f:{−1,1}n→{−1,1} the relationship between the sum of its linear Fourier coefficients f̂(i) (defined by f̂(i)≔12n∑x∈{−1,1}nf(x)xi for i=1,2,…,n and x=(x1,…,xn)) and its degree d is a problem in theoretical computer science related to social choice. In that regard, in 2012, O’Donnell conjectured that ∑i=1nf̂(i)≤dd−1⌊d−12⌋21−d. In 2020, Wang (Wang, 2020) proved that the conjecture is equivalent to a conjecture about cryptographic resilient Boolean functions and proved that the conjecture is true for every n, when d=1 and d=n−1. In this paper, we contribute theoretically to this problem by showing that if the conjecture is true for some specific relatively small value n0, which depends on d, then it is valid for any n≥n0. In particular, we show that the conjecture is correct for every n, when d=2 and d=3. However, despite these promising results, we present a counterexample when d=4, and so the conjecture is not true in general.