Extremal properties of polynomial threshold functions

In this paper we give new extremal bounds on polynomial threshold function (PTF) representations of Boolean functions. Our results include the following: • Almost every Boolean function has PTF degree at most n2 + O( √ n log n). Together with results of Anthony and Alon, this establishes a conjecture of Wang and Williams [32] and Aspnes, Beigel, Furst, and Rudich [4] up to lower order terms. • Every Boolean function has PTF density at most (1 − 1 O(n) )2 . This improves a result of Gotsman [14]. • Every Boolean function has weak PTF density at most o(1)2. This gives a negative answer to a question posed by Saks [28]. • PTF degree blog2 mc+1 is necessary and sufficient for Boolean functions with sparsity m. This answers a question of Beigel [7]. We also give new extremal bounds on polynomials which approximate Boolean functions in the `∞ norm.