A Quantitative Besicovitch 3/4-type Theorem

Below I describe some joint work between myself and my PhD student Matthew Hyde that has been accepted to the journal Annales Academiæ Scientiarum Fennicæ. The paper can be found here.

Similar to how $d$ -dimensional surface measure extends the notion of $d$ -dimensional Lebesgue measure $\mathscr{L}^{d}$ to surfaces, the $d$ –dimensional Hausdorff measure $\mathscr{H}^{d}$ is a generalization of the traditional $d$ -dimensional surface measure that can also be computed for sets that are not surfaces…in fact, they can be computed for any set.

The classical Lebesgue differentiation theorem says that for a measurable subset $A\subseteq\mathbb{R}^{d}$ , then for almost every point $x\in A$ , we have

$\lim_{r\rightarrow 0} \frac{ \mathscr{L}^{d}(A\cap B(x,r))}{\mathscr{L}^{d}(B(x,r))} = 1.$

For $\mathscr{H}^{d}$ measurable sets $A\subseteq \mathbb{R}^{n}$ , this is no longer the case. Certain sets where the Lebesgue differentiation theorem does hold for Hausdorff measure are $d$ –rectifiable sets. These are sets $A$ that can be covered up to $\mathscr{H}^{d}$ -measure zero by countably many Lipschitz images of $\mathbb{R}^{d}$ . You can think of these as generalizations of surfaces, although they no longer need to be smooth or differentiable. Instead of having tangent planes, they have weak tangent planes $\mathscr{H}^{d}$ -a.e. in the sense that at a.e. $x\in A$ , the measure $\mathscr{H}^{d}|_A$ concentrates more and more around a fixed “weak” tangent plane.

A remarkable result of Preiss is a sort of reverse Lebesgue differentiation theorem: if we have a Borel set $A$ so that for a.e. $x\in A$

$\Theta^{d}(A,x):=\lim_{r\rightarrow 0}\frac{\mathscr{H}^{d}(B(x,r)\cap A)}{(2r)^{d}}$

then $A$ is $d$ -rectifiable. In fact, Besicovitch showed that, if $d=1$ , then it is enough to assume that for $\mathscr{H}^{1}$ -a.e. $x\in A$

$\Theta_{*}^{1}(A,x):=\liminf_{r\rightarrow 0}\frac{\mathscr{H}^{1}(B(x,r)\cap A)}{2r}\geq \frac{3}{4}.$

Preiss also shows that $d$ -rectifiability is implied if $\Theta_{*}^{d}(A,x)$ is close enough to $1$ -a.e (the analogous upper density $\Theta^{*,d}(A,x)$ is in fact a.e. at most $1$ for any Borel set $A$ ).

Matthew and I considered the question of whether we could strengthen this result: if we have more information about how large our set was in each ball, could this imply better rectifiable structure, e.g. could we get lower bounds on the largest size of a Lipschitz image intersecting our set?

We showed the following: suppose $A$ is an Ahlfors $d$ -regular set, meaning $\mathscr{H}^{d}(B(x,r)\cap A)\sim r^{d}$ for all $x\in A$ and $0<r<diam \; A$ . Let $\epsilon>0$ and let $\mathscr{B}_{\epsilon}$ be the set of pairs $(x,r)\in A\times (0,r)$ for which there exists $y \in B(x,r)$ and $0<t<r$ satisfying

$\mathscr{H}^{d}_{\infty}(A\cap B(y,t))<(2t)^{d}-\epsilon(2r)^d.$

Here, $\mathscr{H}^{d}_{\infty}$ is Hausdorff content, which is defined as

$\mathscr{H}^{d}_{\infty}(E) = \inf\left\{\sum (diam \; E_i)^{d}: \bigcup E_{i}\supseteq E\right\}.$

If $\mathscr{B}_{\epsilon}$ is a Carleson set for every $\epsilon>0$ , meaning

$\int_{B(x,r)}\int_{0}^{r} \chi_{\mathscr{B}_{\epsilon}}(x,r)\frac{dr}{r}d\mathscr{H}^{d}|_{A}(x)\lesssim_{\epsilon} r^{d}$

then for every ball $B(x,r)$ centered on $A$ there is an $L$ -Lipschitz map $f_{x,r}:\mathbb{R}^{d}\rightarrow \mathbb{R}^{n}$ so that

$\mathscr{H}^{d}(f_{x,r}(B(0,r))\cap A\cap B(x,r))\geq cr^{d}$

where the constants $L$ and $c$ are independent of the choice of ball. In other words, the set $A$ is uniformly rectifiable: not only is the set $A$ covered by Lipschitz graphs, but every ball intersected with our set $A$ contains a big piece of a Lipschitz image with uniform constants.