Query-to-communication lifting using low-discrepancy gadgets

Lifting theorems are theorems that relate the query complexity of a function f : {0, 1}(n) -> {0, 1} to the communication complexity of the composed function f circle g(n) for some "gadget"" g : {0, 1}(b) x {0, 1}(b) -> {0, 1}. Such theorems allow transferring lower bounds from query complexity to the communication complexity, and have seen numerous applications in recent years. In addition, such theorems can be viewed as a strong generalization of a direct-sum theorem for the gadget g. We prove a new lifting theorem that works for all gadgets g that have logarithmic length and exponentially-small discrepancy, for both deterministic and randomized communication complexity. Thus, we significantly increase the range of gadgets for which such lifting theorems hold. Our result has two main motivations: first, allowing a larger variety of gadgets may support more applications. In particular, our work is the first to prove a randomized lifting theorem for logarithmic-size gadgets, thus improving some applications of the theorem. Second, our result can be seen as a strong generalization of a direct-sum theorem for functions with low discrepancy.