Some Connections between Falconer's Distance Set Conjecture and Sets of Furstenburg Type

By Nets Hawk Katz, Terence Tao. 2001

140 citations recorded by the source corpus.

In this paper we investigate three unsolved conjectures in geomet- ric combinatorics, namely Falconer's distance set conjecture, the dimension of Furstenburg sets, and Erdos's ring conjecture. We formulate natural δ- discretized versions of these conjectures and show that in a certain sense that these discretized versions are equivalent.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/d0d6f9254aa6ec981b892b0fd1caccb6153b37ca

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

In Discrete Mathematics and Combinatorics

Citations in the corpus

On Falconer’s distance set problem in the plane (2018, background): “) This set is a variation on an example from [22].”
New Bounds on the Dimensions of Planar Distance Sets (2018, background): “Despite these conceptual differences, we underline that, with the exception of the work of Katz and Tao [13] underpinning Bourgain’s bound (1.”
On self-similar sets with overlaps and inverse theorems for entropy (2012, methodology): “Such a condition appears in the sum-product theorems of Bourgain-Katz-Tao [4] in the work of Katz-Tao [17], and in the Euclidean setting, more explicitly in Bourgain’s work on the Erd˝os-Volkmann conjecture [2] and Marstrand-like projection theorems [3].”
On the Hausdorff dimension of Furstenberg sets and orthogonal projections in the plane (2021, methodology): “Tao [17] connected the 12 -Furstenberg set problem to two other outstanding problems in geometric measure theory: the Falconer distance set problem and the (discretized) ring conjecture.”
On the spectral gap for finitely-generated subgroups of SU(2) (2007, methodology): “Whereas Helfgott used the sum-product theorem in F p, due to Bourgain, Katz, and Tao [4], our proof builds crucially on the metrical counterpart, given by the “discretized ring conjecture” (due to Katz and Tao [ 18 ]), proved by Bourgain [2].”