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In this paper we prove the following results in the plane. They are relatedto 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 isequivalent 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 alsodiscussion about the dependence of $\epsilon_0$ and make a new conjecture.  A tube condition on Frostman measures, after careful refinement, is alsogiven.

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