I will discuss a few applications of the Szemeredi-Trotter type incidence bounds to a few similar sounding questions involving sumsets, such as the convex set sumset problem. These applications yield rather strong, similar looking, and unanimously non-optimal results. It challenges one to try to prove a stronger particular version of the (generally sharp) Szemeredi-Trotter theorem, which can also be interpreted in terms of horocycle flows in SL_2(R).
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