Preprocessed 3SUM for Unknown Universes with Subquadratic Space

We consider the classic 3SUM problem: given sets of integers A, B, C, determine whether there is a tuple (a, b, c) ∈ A × B × C satisfying a + b + c = 0. The 3SUM Hypothesis, central in fine-grained complexity, states that there does not exist a truly subquadratic time 3SUM algorithm. Given this long-standing barrier, recent work over the past decade has explored 3SUM from a data structural perspective. Specifically, in the 3SUM in preprocessed universes regime, we are tasked with preprocessing sets A, B of size n, to create a space-efficient data structure that can quickly answer queries, each of which is a 3SUM problem of the form A', B', C', where A' ⊆ A and B' ⊆ B. A series of results have achieved Õ(n^2) preprocessing time, Õ(n^2) space, and query time improving progressively from Õ(n^1.9) [CL15] to Õ(n^11/6) [CVX23] to Õ(n^1.5) [KPS25]. Given these series of works improving query time, a natural open question has emerged: can one achieve both truly subquadratic space and truly subquadratic query time for 3SUM in preprocessed universes? We resolve this question affirmatively, presenting a tradeoff curve between query and space complexity. Specifically, we present a simple randomized algorithm achieving Õ(n^1.5 + ε) query time and Õ(n^2 - 2ε/3) space complexity. Furthermore, our algorithm has Õ(n^2) preprocessing time, matching past work. Notably, quadratic preprocessing is likely necessary for our tradeoff as either the preprocessing or the query time must be at least n^2-o(1) under the 3SUM Hypothesis.