Counting Short Vector Pairs by Inner Product and Relations to the Permanent

Given as input two n-element sets A, B ⊆ {0,1}^d with d = clog n ≤ (log n)²/(log log n)⁴ and a target t ∈ {0,1,…,d}, we show how to count the number of pairs (x,y) ∈ A× B with integer inner product ⟨ x,y ⟩ = t deterministically, in n²/2^{Ω(√{log nlog log n/(clog² c)})} time. This demonstrates that one can solve this problem in deterministic subquadratic time almost up to log² n dimensions, nearly matching the dimension bound of a subquadratic randomized detection algorithm of Alman and Williams [FOCS 2015]. We also show how to modify their randomized algorithm to count the pairs w.h.p., to obtain a fast randomized algorithm. Our deterministic algorithm builds on a novel technique of reconstructing a function from sum-aggregates by prime residues, or modular tomography, which can be seen as an additive analog of the Chinese Remainder Theorem. As our second contribution, we relate the fine-grained complexity of the task of counting of vector pairs by inner product to the task of computing a zero-one matrix permanent over the integers.