Abstract: The Favard length of a compact set is the average length of its orthogonal projections onto one-dimensional linear subspaces. A classical result known as the Besicovitch–Federer projection theorem states that this average vanishes for purely unrectifiable sets of finite length. A more quantitative variant asks: how quickly does the Favard length of small neighbourhoods of such a set tends to zero?

 

In this talk, we discuss power-law decay estimates for self-similar product Cantor sets in arbitrary ambient dimension, including improved estimates for the classical four-corner Cantor set in the plane. We will begin with the classical projection theory underlying the problem, including Federer’s theorem on coordinate projections and its role in identifying purely unrectifiable examples.

 

The main focus of the talk will be the interplay between geometry, harmonic analysis, and combinatorics. Fourier analysis shows that most directions in the average exhibit substantial projection overlap; whereas, a combinatorial argument propagates this overlap (via self-similarity of the Cantor set) to finer scales. Together, these two arguments then force the average projected length to become small.

 

I will sketch refinements of the combinatorial argument that improve the resulting decay estimates. Then, as a concrete illustration, we apply our general result to Cantor sets whose digit-set polynomials have no zeros on the unit circle. These examples simplify the Fourier analysis and make the contribution of the combinatorial improvements particularly explicit.

 

Host: Alan Chang