Query-to-Communication Lifting for BPP

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“…Such lower bounds fall short of the current best record of exp(N 1/3−o(1) ) due to Harnik and Raz [25]. We inherit the need for large gadgets from prior work [19,22]; see Section 4. For this reason (and others), it is an important open problem to develop a lifting theory for gadgets of size m = O(1).…”

“…Techniques. We use tools developed in the context of tree-like lifting theorems, specifically from [19,22]. These tools allow us to relate large rectangles in the input domain of S • IND n m with large subcubes in the input domain of S; see Section 4.…”

“…In this section, as preparation, we recall some technical notions from [19,22] concerning the index gadget g := IND m . Specifically, writing G := g n : [m] n × {0, 1} mn → {0, 1} n for n copies of g, we explain how large rectangles in the domain of G are related with large subcubes in the codomain of G. In what follows, we will always assume that m ≥ n ∆ for a sufficiently large constant ∆.…”

“…For a set X we use the boldface X X X to denote a random variable uniformly distributed over X. 17,22]). For m ≥ n ∆ , every ρ-structured rectangle is ρ-like.…”

“…We claim that, given any rectangle R := X ×Y ⊆ [m] n × {0, 1} mn , we can partition most of X ×Y into ρ-structured subrectangles with |fix ρ| bounded in terms of the size of X × Y . Indeed, we describe a simple 2-round partitioning scheme from [22] below; see also…”