Asian Journal of Mathematics

Volume 27 (2023)

Number 6

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

Pages: 829 – 852

DOI: https://dx.doi.org/10.4310/AJM.2023.v27.n6.a1

Authors

Bochen Liu (Department of Mathematics & International Center for Mathematics, Southern University of)

Chun-Yen Shen (Department of Mathematics, National Taiwan University, and National Center for TheoreticalSciences, Taipei, Taiwan)

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 a nontrivial exponent on intersection between pencils of $\delta$-tubes, under nonconcentration conditions. In fact we show it is equivalent to the discretized sum-product problem.

Then we use our estimates of pencils to prove new results on dimensions of radial projections. We also make a conjecture that would reveal a key difference between orthogonal projections and radial projections.

Keywords

radial projection, pencil of tubes, discretized sum-product, tube condition

2010 Mathematics Subject Classification

11B30, 28A75

Received 28 September 2022

Accepted 2 August 2023

Published 7 August 2024