Favard length estimates via cyclotomic divisibility

Abstract: The Favard length of a planar set $E$ is the average length of its one-dimensional projections. It is well known (due to Besicovitch) that if $E$ is a purely unrectifiable planar self-similar set of Hausdorff dimension 1, then its Favard length is 0. Consequently, if $E_\delta$ is the $\delta$-neighbourhood of $E$, then the Favard length of $E_\delta$ goes to 0 as $\delta\to 0$. A question of interest in geometric measure theory, ergodic theory and analytic function theory is to estimate the rate of decay, both from above and below. Partial results in this direction have been proved by many authors, including Mattila, Nazarov, Perez, Volberg, Bond, Bateman, and myself. In addition to geometric measure theory, this work has involved methods from harmonic analysis, additive combinatorics, and algebraic number theory. I will review the relevant background, and then discuss my recent work with Caleb Marshall on upper bounds on the Favard length for 1-dimensional planar Cantor sets with a rational product structure. This improves on my earlier work with Bond and Volberg, and incorporates new methods introduced in my work with Itay Londner on integer tilings.

classical analysis and ODEsfunctional analysisrepresentation theoryspectral theory

Audience: researchers in the topic

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