Reading group on the Falconer distance conjecture

Where and when : Friday 2-3pm, Math 126.

This reading seminar will focus on the Falconer distance conjecture.

If E ⊂ ℝ², define D(E) = { |x-y| : x, y ∈ E },
where |x-y| is the Euclidean distance between the points x and y.

Falconer conjectured that if the set E is big, then the set D(E) must also be big. More precisely, if E has Hausdorff dimension larger than 1, then D(E) must have positive Lebesgue measure. The problem is closely related to the Erdős distinct distances problem, and also to several questions in geometric measure theory.

This reading seminar will focus on two papers. The first is by Tuomas Orponen, who proved that the Falconer distance conjecture is true for all sets E That are Ahlfors-David regular. Orponen's paper can be found here. The second paper is by Nets Katz and Terry Tao, who showed that the Falconer distance conjecture is related to several other problems in harmonic analysis and additive combinatorics, such and the Kakeya problem and sum-product theory. Katz and Tao's paper can be found here.

Schedule

Announcements