Settling The Query Complexity Of Non-Adaptive Junta Testing

We prove that any non-adaptive algorithm that tests whether an unknown Boolean function f : {0, 1}(n) -> {0, 1} is a k-junta or epsilon-far from every k-junta must make (Omega) over tilde (k(3/2)/epsilon) many queries for a wide range of parameters k and epsilon. Our result dramatically improves previous lower bounds from [12, 38], and is essentially optimal given Blais's non-adaptive junta tester from [7], which makes (O) over tilde (k(3/2))/epsilon queries. Combined with the adaptive tester of [8] which makes O(k log k + k/epsilon) queries, our result shows that adaptivity enables polynomial savings in query complexity for junta testing.