A polynomial restriction lemma with applications
Valentine Kabanets, Daniel M. Kane, Zhenjian Lu · 2017
A polynomial threshold function (PTF) of degree d is a boolean function of the form f=sgn(p), where p is a degree-d polynomial, and sgn is the sign function. The main result of the paper is an almost optimal bound on the probability that a random restriction of a PTF is not close to a constant function, where a boolean function g is called δ-close to constant if, for some vε{1,-1}, we have g(x)=v for all but at most δ fraction of inputs. We show for every PTF f of degree d≥ 1, and parameters 0<δ, r≤ 1/16, that