The Correct Exponent for the Gotsman-Linial Conjecture

Abstract: We prove a new bound on the average sensitivity of polynomial threshold functions. In particular we show that a polynomial threshold function of degree $d$ in at most $n$ variables has average sensitivity at most $$\sqrt{n}(\log(n))<sup>{O(d\log(d))}2<sup>{O(d<sup>2\log(d)}$$. For fixed $d$ the exponent in terms of $n$ in this bound is known to be optimal. This bound makes significant progress towards the Gotsman-Linial Conjecture which would put the correct bound at $\Theta(d\sqrt{n})$.