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.

References in corpus (7)

• Some connections between Falconer's distance set conjecture, and sets of Furstenburg type
• On the dimension and smoothness of radial projections
• New bounds on the dimensions of planar distance sets
• A sharp exceptional set estimate for visibility
• On the discretized sum-product problem
• Orthogonal projections of discretized sets
• On Hausdorff dimension of radial projections

Cited by in corpus (1)

• Planar incidences and geometric inequalities in the Heisenberg group