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The Favard Length and Cyclotomic Structure of Rational Product Cantor Sets

Caleb Marshall (UBC)
Tuesday, March 26, 2024 - 1:30pm to 3:30pm
PDL C-401

The Favard Length of a compact subset of the plane is the average length of its projection onto linear subspaces. Due to a theorem of A. Besicovitch, the Favard length of a set of finite length is zero if-and-only-if the set is purely 1-unrectifiable. However, precise asymptotics for delta thickenings (as delta decreases to zero) of unrectifiable sets is still an open question--even over families of sets with many additional structures. We call such asymptotic statements Quantitative Besicovitch Projection Theorems (QBPT's).

In the first portion of this talk, we discuss QBPT's for rational product Cantor sets--a class of self-similar product sets whose iterated function systems are given by rational digit sets. We motivate these sets as, ``canonical working examples'', for studying the Favard Length and QBPT's by contrasting their projection's behaviour with those of less-structured sets. We also discuss some foundational tools--such as Riesz products and Mask Polynomials of Digit Sets--which we use to prove QBPT's for rational product Cantor sets. Emphasis is placed on our new advances in estimating Riesz products associated to these sets.In the second portion of the talk, we focus on the Cyclotomic Structure of digit sets. This introduces another of our new advancements--an extension of the Lam-Leung structure theorem for cyclotomic polynomial divisors. We sketch the connection between such structure theorems for cyclotomic polynomials, strong estimates upon associated Riesz products, and QBPT's for rational product Cantor sets. We conclude with ongoing research and some open questions.This is joint work with Izabella Łaba (UBC)

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