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

Abstract: 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 $\epsilon_0$-increment on intersection between pencils of $\delta$-tubes, under non-concentration conditions. In fact we show it is equivalent to the discretized sum-product problem, thus the $\epsilon_0$ follows from Bourgain's celebrated result. Then we prove a couple of new results on radial projections. We also discussion about the dependence of $\epsilon_0$ and make a new conjecture. A tube condition on Frostman measures, after careful refinement, is also given.