A theorem of Elekes and Szabó recognizes algebraic groups among certain complex algebraic varieties with maximal size intersections with finite grids. We establish a generalization to relations of any arity and dimension definable in: (1) stable structures with distal expansions (includes algebraically and differentially closed fields of characteristic 0) and (2) o-minimal expansions of groups. Our methods provide explicit bounds on the power-saving exponent in the nongroup case. Ingredients of the proof include a higher arity generalization of the abelian group configuration theorem in stable structures (along with a purely combinatorial variant characterizing Latin hypercubes that arise from abelian groups) and Zarankiewicz-style bounds for hypergraphs definable in distal structures.

We study the dimension of self-similar measures associated to a homogeneous iterated function system of three contracting similarities on R and other more general iterated function systems. We extend some of the theory recently developed for Bernoulli convolutions to this setting. In the setting of three maps a new phenomenon occurs, which has been highlighted by recent examples of Baker as well as Bárány and Käenmäki. To overcome the difficulties stemming from this phenomenon, we develop novel techniques, including an extension of Hochman’s entropy increase method to a function field setup.