A Quantification of a Besicovitch Non-linear Projection Theorem via Multiscale Analysis

Abstract

The Besicovitch projection theorem states that if a subset E of the plane has finite length in the sense of Hausdorff measure and is purely unrectifiable (so its intersection with any Lipschitz graph has zero length), then almost every orthogonal projection of E to a line will have zero measure. In other words, the Favard length of a purely unrectifiable 1-set vanishes. In this article, we show that when linear projections are replaced by certain non-linear projections called curve projections, this result remains true. In fact, we go further and use multiscale analysis to prove a quantitative version of this Besicovitch non-linear projection theorem. Roughly speaking, we show that if a subset of the plane has finite length in the sense of Hausdorff and is nearly purely unrectifiable, then its Favard curve length is very small. Our techniques build on those of Tao, who in (Proc Lond Math Soc 98:559-584, 2009) proves a quantification of the original Besicovitch projection theorem.

Dimension and measure of sums of planar sets and curves

“…We note that a deeper investigation of the result of Corollary 2.6 is followed up on in [2, 3], where we obtain upper and lower bounds on the rate of decay as $$n\rightarrow \infty$$ of the probability that a unit circle intersects the n th generation in the construction of the four corner Cantor set, $$C(1/4)$$. Further, in [2, 4], we obtain similar quantitative information for more general irregular 1‐sets.…”

“…We extend the curve Γ and introduce extension operators so that $$\Phi _\lambda (a)$$ is defined for each $$\lambda \in [0,2L]$$ and each $$a\in A$$. To this end, extend the function γ on I to a function $$\widetilde{\gamma }$$ on $$\widetilde{I}=[ -\frac{L}{2},2L ]$$ so that $$\begin{equation*} \widetilde{\Gamma }= \lbrace (t, \widetilde{\gamma }(t)) : t\in \widetilde{I} \rbrace \end{equation*}$$is a good curve in the sense of Definition 4.1; The details of this extension are given in [4, Section 2.3]. For $$x = (x_1, x_2) \in \Omega =[0, \frac{L}{2}]^2$$ and $$\lambda \in [0,2L]$$, define $$\begin{equation} \widetilde{\Phi }_\lambda (x) ...$$…”

“…This work is continued in [31], where we investigate the existence of an interior point in sum sets of the form $$A+\Gamma$$. It is also continued in [2–4], where more quantitative estimates are obtained, and in [23] where pinned distance sets and paths are investigated in the more specific setting of Cantor sets.…”