The Dual BKR Inequality and Rudich's Conjecture

Let T be a set of terms over an arbitrary (but finite) number of Boolean variables. Let U(T ) be the set of truth assignments that satisfy exactly one term in T . Motivated by questions in computational complexity, Rudich conjectured that there exist , δ > 0 such that if T is any set of terms for which U(T ) contains at least a (1− )-fraction of all truth assignments, then there exists a term t ∈ T such that at least a δ-fraction of assignments satisfy some term of T sharing a variable with t [7]. We prove a stronger version: for any independent assignment of the variables (not necessarily the uniform one), if the measure of U(T ) ∗Supported in part by NSF grants DMS0200856 and DMS0701175. †Supported in part by NSF Grant CCR-9700239 and CCR-0832787.