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We present a collection of new results on problems related to 3SUM,including:  1. The first truly subquadratic algorithm for  $\ \ \ \ \ $ 1a. computing the (min,+) convolution for monotone increasingsequences with integer values bounded by $O(n)$,  $\ \ \ \ \ $1b. solving 3SUM for monotone sets in 2D with integer coordinatesbounded by $O(n)$, and  $\ \ \ \ \ $1c. preprocessing a binary string for histogram indexing (alsocalled jumbled indexing).  The running time is:$O(n^{(9+\sqrt{177})/12}\,\textrm{polylog}\,n)=O(n^{1.859})$ withrandomization, or $O(n^{1.864})$ deterministically. This greatly improves theprevious $n^2/2^{\Omega(\sqrt{\log n})}$ time bound obtained from Williams'recent result on all-pairs shortest paths [STOC'14], and answers an openquestion raised by several researchers studying the histogram indexing problem.  2. The first algorithm for histogram indexing for any constant alphabet sizethat achieves truly subquadratic preprocessing time and truly sublinear querytime.  3. A truly subquadratic algorithm for integer 3SUM in the case when the givenset can be partitioned into $n^{1-\delta}$ clusters each covered by an intervalof length $n$, for any constant $\delta>0$.  4. An algorithm to preprocess any set of $n$ integers so that subsequently3SUM on any given subset can be solved in $O(n^{13/7}\,\textrm{polylog}\,n)$time.  All these results are obtained by a surprising new technique, based on theBalog--Szemer\'edi--Gowers Theorem from additive combinatorics.

Clustered Integer 3SUM via Additive Combinatorics