Tangency counting for well-spaced circles

2025/04/19 by Maldague, Dominique, Ortiz, Alexander
#42B10 #42B25 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2504.14118

Abstract

In the late 90's, Tom Wolff introduced the circle tangency counting problem in his expository article on the Kakeya conjecture. For collections of well-spaced circles, we break the N^3/2-barrier, proving that a set of N well-spaced circles has at most N^25/18+ε sites of internal tangency. The circle tangency problem can be related to a problem about incidences between points in ℝ³ and light rays. For this problem, we introduce a stopping time argument to extract maximal information about well-spaced points from a refined decoupling theorem for the light cone in ℝ³, leading to sharp bounds on the number of μ-rich tangency rectangles.

Related

• A solution to Ringel's circle problem
• Depth of segments and circles through points enclosing many points: a note
• A conditional bound on sphere tangencies in all dimensions
• Some Problems in Number Theory I: The Circle Problem
• Inscribed rectangles in a smooth Jordan curve attain at least one third\n of all aspect ratios