Relationships between the number of inputs and other complexity measures of Boolean functions

Abstract

We generalize and extend the ideas in a recent paper of Chiarelli, Hatami and Saks to prove new bounds on the number of relevant variables for boolean functions in terms of a variety of complexity measures. Our approach unifies and refines all previously known bounds of this type. We also improve Nisan and Szegedy's well-known inequality bs(f ) ≤ deg(f ) 2 by a constant factor, thereby improving Huang's recent proof of the sensitivity conjecture by the same constant.

Junta Threshold for Low Degree Boolean Functions on the Slice

“…The result proved in [FI19a] is in fact stronger: under the same assumptions, there is a Boolean degree d function g on the Boolean cube {0, 1} n such that f is the restriction of g to the slice. This implies the junta conclusion since every Boolean degree d function on the Boolean cube is an O(2 d )-junta [NS94,CHS20,Wel20].…”

“…Chiarelli, Hatami and Saks [CHS20] improved the bound to O(2 d ), and the hidden constant was further optimized by Wellens [Wel20]. The slice…”