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 = c log 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 n2/2Ω(√log n log log n/(c log2 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.