We study the equivalence testing problem where the goal is to determine if the given two unknown distributions on [n] are equal or far in the total variation distance in the conditional sampling model wherein a tester can get a sample from the distribution conditioned on any subset. Equivalence testing is a central problem in distribution testing, and there has been a plethora of work on this topic in various sampling models.
Despite significant efforts over the years, there remains a gap in the current best-known upper bound of O(log log n) [FJOPS, COLT 2015] and lower bound of \Omega(\sqrt{log log n}) [ACK, Theory of Computing 2018]. Closing this gap has been repeatedly posed as an open problem (listed as open problem 87 at sublinear.info). In this work, we completely resolve the query complexity of this problem by showing a lower bound of \Omega(log log n). For that purpose, we develop a novel and generic proof technique that enables us to break the \sqrt{log log n} barrier, not only for the equivalence testing problem but also for other distribution testing problems.

(joint work with Sourav Chakraborty and Gunjan Kumar)