The Favard length problem for self-similar sets

The Favard length of a Borel set E ⊂ ℝᵈ is the average value of the one-dimensional Hausdorff measure (i.e. length) of the orthogonal projection of E onto one-dimensional linear subspaces. We denote this average projected length by Fav(E). Due to the work of Besicovitch and Federer, a set E ⊂ ℝᵈ having positive and finite length satisfies Fav(E) = 0 if and only if E intersects all Lipschitz graphs in a set of null length. Such sets are called (purely 1)-unrectifiable sets. A quantitative reformulation of this result is as follows. Let δ > 0, and suppose that Eδ ⊂ ℝᵈ : (0,1) → (0,∞) such that Fav(Eδ) ≤ ψE(δ) for all δ ∈ (0,1) and lim(δ→0⁺) ψE(δ) = 0. The Favard length problem asks: what is the optimal choice of function ψE? Since 2001, the Favard length problem has seen significant attention within the field of mathematics known as geometric measure theory. For a class of unrectifiable sets in the plane known as rational product Cantor sets, there are strong asymptotic upper bounds (such as those of F. Nazarov, Y. Peres and A. Volberg) which are close to known sharp lower bounds (due to P. Mattila). These upper bounds rely upon Fourier analytic estimates, as well as combinatorial estimates, which demonstrate an emerging connection between the Favard length problem and size bounds for polynomials with many cyclotomic polynomial divisors. This dissertation presents the author’s work on the Favard length problem, with the main result being that the known planar upper bound for rational product Cantor sets in ℝ² generalizes completely to a class of rational product Cantor sets in ℝᵈ for d ≥ 3. The connection between the Favard length problem and cyclotomic divisibility is also explored in more detail. In particular, an explicit connection between the Favard length problem and the author’s recent size bounds for mask polynomials with many cyclotomic divisors is established.