Preprocessed 3SUM: From Reals to Integers

Abstract: The Preprocessed 3SUM problem asks to preprocess a set $A$ of $n$ numbers so that we can quickly decide whether a query subset $A&#39; \subseteq A$ contains a solution $a+b=c$. The best known query bounds differ substantially between integer and real inputs. For integers, a series of advances with near-quadratic preprocessing reduced the query time from $\tilde O(n<sup>{13/7})$ [Chan, Lewenstein; STOC '15] to $\tilde O(n<sup>{11/6})$ [Chan, Vassilevska W., Xu; STOC '23], and finally to $\tilde O(n<sup>{3/2})$ [Kasliwal, Polak, Sharma; SOSA '25]. For reals, only the first of these bounds has been achieved, by adapting the corresponding integer algorithm [Fischer; SODA '25]. Is this gap inherent to the more limited tools available for real inputs? We give a randomized real-to-integer reduction with $\tilde O(n<sup>2)$ preprocessing overhead and no asymptotic increase in query time. This yields $\tilde O(n<sup>2)$ preprocessing and $\tilde O(n<sup>{3/2})$ query time for real inputs, closing the gap. We show how to extend our result to the setting where one set is unknown at preprocessing time. Our reduction is conceptually simple, and relies on an interesting technical toolkit around old and new tools from sparse linear algebra. More broadly, it suggests a route to real-to-integer reductions for other additive problems.