On variants of the Furstenberg set problem
Abstract
Given an integer d ⩾ 2 d \geqslant 2 , s ∈ ( 0 , 1 ] s \in (0,1] , and t ∈ [ 0 , 2 ( d − 1 ) ] t \in [0,2(d-1)] , suppose a set X X in R d \mathbb {R}^d has the following property: there is a collection of lines of packing dimension t t such that every line from the collection intersects X X in a set of packing dimension at least s s . We show that such sets must have packing dimension at least max { s , t / 2 } \max \{s,t/2\} and that this bound is sharp. In particular, the special case d = 2 d=2 solves a variant of the Furstenberg set problem for packing dimension. We also solve the upper and lower box dimension variants of the problem. In both of these cases the sharp threshold is max { s , t + 1 − d } \max \{s,t+1-d\} .
Fields of Study & Subjects
Mathematical Dynamics and Fractals
Advanced Harmonic Analysis Research
Limits and Structures in Graph Theory
Publication details
- Journal / source
- Proceedings of the American Mathematical Society
- Volume
- 154
- Issue
- 3
- Pages
- 1093–1103
- Publication date
- 14 January 2026
- Language
- EN