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Abstract incidence geometries like projective spaces, polar spaces, generalized polygons, as well as incidence problems in the real or complex Euclidean spaces (eg. Szemerédi–Trotter theorem).
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Is the sumset or the sumset of the square set always large?
Six years later it's hard to know if this is still a question of interest. For the record (since I stumbled across the question), the answer is yes and it's a consequence of a more general phenomenon. …