Intersection between pencils of tubes, discretized sum-product, and radial projections

In this paper we prove the following results in the plane. They are related to each other, while each of them has its own interest. First we obtain an $ε_0$-increment on intersection between pencils of $δ$-tubes, under non-concentration conditions. In fact we show it is equivalent to the discretized sum-product problem, thus the $ε_0$ follows from Bourgain's celebrated result. Then we prove a couple of new results on radial projections. We also discussion about the dependence of $ε_0$ and make a new conjecture. A tube condition on Frostman measures, after careful refinement, is also given.

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