Exponentially small soundness for the direct product z-test

Given a function f : [N]k ! [M]k, the Z-test is a three query test for checking if a function f is a direct product, namely if there are functions g1, . . . gk : [N] ! [M] such that f(x1, . . . , xk) = (g1(x1), . . . gk(xk)) for every input x 2 [N]k. This test was introduced by Impagliazzo et. al. (SICOMP 2012), who showed that if the test passes with probability ϵ > exp(- p k) then f is (ϵ) close to a direct product function in some precise sense. It remained an open question whether the soundness of this test can be pushed all the way down to exp(-k) (which would be optimal). This is our main result: we show that whenever f passes the Z test with probability ϵ > exp(-k), there must be a global reason for this: namely, f must be close to a product function on some (ϵ) fraction of its domain. Towards proving our result we analyze the related (two-query) V-test, and prove a "restricted global structure" theorem for it. Such theorems were also proven in previous works on direct product testing in the small soundness regime. The most recent work, by Dinur and Steurer (CCC 2014), analyzed the V test in the exponentially small soundness regime. We strengthen their conclusion of that theorem by moving from an "in expectation" statement to a stronger "concentration of measure" type of statement, which we prove using hyper-contractivity. This stronger statement allows us to proceed to analyze the Z test. We analyze two variants of direct product tests. One for functions on ordered tuples, as above, and another for functions on sets, f : [N] k ! [M]k. The work of Impagliazzo et. al was actually focused only on functions of the latter type, i.e. on sets. We prove exponentially small soundness for the Z-test for both variants. Although the two appear very similar, the analysis for tuples is more tricky and requires some additional ideas.