Scenery flow, conical densities, and rectifiability

Measure preserving flow

Ergodic theory studies the asymptotic behaviour of typical orbits of dynamical systems endowed with an invariant measure. Geometric measure theory can be described as a field of mathematics where geometric problems on sets and measures are studied via measure-theoretic techniques. Our main innovation is to apply the scenery flows to study classical problems in geometric measure theory which a priori do not involve any dynamics. The idea is to study the structure of a measure μ\mu on Rd\mathbb{R}^d via dynamical properties of its magnifications at a given point xRdx \in \mathbb{R}^d. Rather than individual results, we believe that our main contribution is to highlight the relevance of ergodic-theoretic methods around the scenery flow in geometric problems. Let XX be a metric space and write R+=[0,)\mathbb{R}_+ = [0,\infty). A one-sided flow is a family (St)tR+(S_t)_{t \in \mathbb{R}_+} of maps St ⁣:XXS_t \colon X \to X for which

St+t=StSt,t,tR+.S_{t+t'} = S_{t} \circ S_{t'}, \qquad t,t' \in \mathbb{R}_+.

In other words, (St)(S_t) is an additive R+\mathbb{R}_+ action on XX. If (X,B,P)(X,\mathcal{B},P) is a probability space, then we say that PP is StS_t invariant if StP=PS_t P = P for all t0t \geq 0. In this case, we call (X,B,P,(St)tR+)(X,\mathcal{B},P,(S_t)_{t \in \mathbb{R}_+}) a measure preserving flow. A set ABA \in \mathcal{B} is StS_t invariant if P(St1AA)=0P(S_t^{-1} A \triangle A)=0. A measure preserving flow is ergodic, if for all t0t \geq 0 the measure PP is ergodic with respect to the transformation St ⁣:XXS_t \colon X \to X, that is, for all StS_t invariant sets ABA \in \mathcal{B} we have P(A){0,1}P(A) \in \{ 0,1 \}.

Theorem 1 (Birkhoff ergodic theorem). If (X,B,P,(St)tR+)(X,\mathcal{B},P,(S_t)_{t \in \mathbb{R}_+}) is an ergodic measure preserving flow, then for a PP integrable function f ⁣:XRf \colon X \to \mathbb{R} we have

limT1T0Tf(Stx)dt=fdP\lim_{T \to \infty} \frac{1}{T} \int_0^T f(S_t x) \,\mathrm{d} t = \int f \,\mathrm{d} P

at PP almost every xXx \in X.

Theorem 2 (Ergodic decomposition). Any StS_t invariant measure PP can be decomposed into ergodic components PωP_\omega, ωP\omega \sim P, such that

P=PωdP(ω).P = \int P_\omega \,\mathrm{d} P(\omega).

Ergodic decomposition is unique up to PP-measure zero sets.

Scenery flow

Let M1=P(B1)\mathcal{M}_1 = \mathcal{P}(B_1) be the collection of all Borel probability measures on the unit ball B1B_1 and M1={μM1:0sptμ}\mathcal{M}_1^* = \{ \mu \in \mathcal{M}_1 : 0 \in \mathrm{spt}\mu \}. Define the magnification StμS_t\mu of μM1\mu \in \mathcal{M}_1^* at 00 by

Stμ(A)=μ(etA)μ(B(0,et)),AB1.S_t\mu(A) = \frac{\mu(e^{-t}A)}{\mu(B(0,e^{-t}))}, \qquad A \subset B_1.

Due to the exponential scaling, (St)tR+(S_t)_{t \in \mathbb{R}_+} is a flow in the space M1\mathcal{M}_1^* and we call it the scenery flow at 00. Although the action StS_t is discontinuous (at measures μ\mu with μ(B1)>0\mu(\partial B_1) > 0) and the space M1M1\mathcal{M}_1^* \subset \mathcal{M}_1 is not closed (but it is Borel), the scenery flow behaves in a very similar way to a continuous flow on a compact metric space. If we have a Radon measure μ\mu and xsptμx \in \mathrm{spt}\mu, we want to consider the scaling dynamics when magnifying around xx. Let Txμ(A)=μ(A+x)T_x\mu(A) = \mu(A+x) and define μx,t=St(Txμ)\mu_{x,t} = S_t(T_x\mu). Then the one-parameter family (μx,t)tR+(\mu_{x,t})_{t \in \mathbb{R}_+} is called the scenery flow at xx. Accumulation points of this scenery in M1\mathcal{M}_1 will be called tangent measures of μ\mu at xx and the family of tangent measures of μ\mu at xx is denoted by Tan(μ,x)M1\mathrm{Tan}(\mu,x) \subset \mathcal{M}_1. We are not interested in a single tangent measure, but the whole statistics of the scenery μx,t\mu_{x,t} as tt \to \infty, i.e. the tangent distribution. The tangent distribution of μ\mu at xsptμx \in \mathrm{spt}\mu is any weak limit of

μx,T=1T0Tδμx,tdt.\langle \mu \rangle_{x,T} = \frac{1}{T} \int_0^T \delta_{\mu_{x,t}} \,\mathrm{d} t.

The family of tangent distributions of μ\mu at xx is denoted by TD(μ,x)P(M1)\mathcal{TD}(\mu,x) \subset \mathcal{P}(\mathcal{M}_1). The integration above makes sense since we are on a convex subset of a topological linear space. We emphasize that tangent distributions are measures on measures. If the limit above is unique, then, intuitively, it means that the collection of views μx,t\mu_{x,t} will have well defined statistics when zooming into smaller and smaller neighbourhoods of xx. Notice that the set TD(μ,x)\mathcal{TD}(\mu,x) is non-empty and compact at xsptμx \in \mathrm{spt}\mu. Moreover, the support of each PTD(μ,x)P \in \mathcal{TD}(\mu,x) is contained in Tan(μ,x)\mathrm{Tan}(\mu,x).

Fractal distributions

We say that the distribution PP on M1\mathcal{M}_1 is scale invariant if it is StS_t invariant, that is, StP=PS_tP = P for all t0t \ge 0, and quasi-Palm if for any Borel set AM1\mathcal{A} \subset \mathcal{M}_1 with P(A)=1P(\mathcal{A}) = 1 it holds that PP almost every νA\nu \in \mathcal{A} satisfies

νz,tA\nu_{z,t} \in \mathcal{A}

for ν\nu almost every zRdz \in \mathbb{R}^d with B(z,et)B1B(z,e^{-t}) \subset B_1. Roughly speaking, the quasi-Palm property guarantees that the null sets of the distributions are invariant under translations to a typical point of the measure. The distribution PP on M1\mathcal{M}_1 is a fractal distribution (FD) if it is scale invariant and quasi-Palm. A fractal distribution is an ergodic fractal distribution (EFD) if it is ergodic with respect to StS_t. Write FD\mathcal{FD} and EFD\mathcal{EFD} for the set of all fractal distributions and ergodic fractal distributions, respectively. A general principle is that tangent objects enjoy some kind of spatial invariance. For example, Preiss (1987) proved that tangent measures to tangent measures are tangent measures. For tangent distributions, a very powerful formulation of this principle is the following theorem.

Theorem 3 (Hochman (2010+)). For any Radon measure μ\mu and μ\mu almost every xx, all tangent distributions at xsptμx \in \mathrm{spt}\mu are fractal distributions.

Notice that as the action StS_t is discontinuous, even the scale invariance of tangent distributions or the fact that they are supported on M1\mathcal{M}_1^* are not immediate, though they are perhaps expected. The most interesting part in the above theorem is that a typical tangent distribution satisfies the quasi-Palm property. Hochman’s result is highly nontrivial. It is proved by using CP processes which are Markov processes on the dyadic scaling sceneries of a measure introduced by Furstenberg. Technical difficulties of the proof include an interplay between fractal distributions and CP processes, restricted and extended versions of distributions, and different norms. Although fractal distributions are defined in terms of seemingly strong geometric properties, the family of fractal distributions is in fact very robust.

Theorem 4 (K-Sahlsten-Shmerkin (2015)). The family of fractal distributions is compact.

The result may appear rather surprising since the scenery flow is not continuous, its support is not closed, and, more significantly, the quasi-Palm property is not a closed property. The proof of this result is also based on the interplay between fractal distributions and CP processes, and restricted and extended versions of distributions. Together with convexity, the theorem implies that the family of fractal distributions is in fact a Choquet simplex.

Theorem 5 (K-Sahlsten-Shmerkin (2015)). The family of fractal distributions is a Poulsen simplex, i.e. a Choquet simplex in which extremal points are dense.

Note that the set of extremal points is precisely the collection of ergodic fractal distributions. The proof of this result is again based on the interplay between fractal distributions and CP processes. We prove that ergodic CP processes are dense by constructing a dense set of distributions of random self-similar measures on the dyadic grid. This is done by first approximating a given CP process by a finite convex combination of ergodic CP processes, and then, by splicing together those finite ergodic CP processes, constructing a sequence of ergodic CP processes converging to the convex combination. Roughly speaking, splicing of measures consists in pasting together a sequence of measures along dyadic scales. Splicing is often employed to construct measures with a given property based on properties of the component measures.

This idea was used in the movie Splice in 2009 where scientists formed new species by splicing together DNA of different animals. Schmeling-Shmerkin (2010) used the idea to investigate the dimensions of iterated sums of Cantor sets and Hochman (2010+) employed it to construct certain examples.

Uniformly scaling measures

In geometric considerations, we usually construct a fractal distribution satisfying certain property. We often want to transfer that property back to a measure. This leads us to the concept of generated distributions. We say that a measure μ\mu generates a distribution PP at xx if

TD(μ,x)={P}.\mathcal{TD}(\mu,x) = \{ P \}.

Furthermore, μ\mu is a uniformly scaling measure (USM) if μ\mu generates PP at μ\mu almost every xx. One can think that uniformly scaling property is an ergodic-theoretical notion of self-similarity. Hochman (2010+) proved the striking fact that generated distributions are always fractal distributions. The following result is a converse to this.

Theorem 6 (K-Sahlsten-Shmerkin (2015)). For any fractal distribution PP, there exists a uniformly scaling measure μ\mu generating PP.

By the compactness and the Poulsen property (by the Birkhoff ergodic theorem, the claim holds for ergodic fractal distributions), it suffices to show that the collection of fractal distributions satisfying the claim is closed. The proof of this is again based on the interplay between fractal distributions and CP processes. Even though the previous theorem concerns measures which have a single tangent distribution at typical points, perhaps surprisingly, it gives us an application which shows that the exact opposite holds for a Baire generic measure.

Theorem 7 (K-Sahlsten-Shmerkin (2015)). For a Baire generic Radon measure μ\mu, the set of tangent distributions is the set of all fractal distributions at μ\mu almost every xx.

The result continues the work of O’Neil (1994) and Sahlsten (2014) who proved that a Baire generic measure has all Borel measures as tangent measures at almost every point.

Dimension of fractal distributions

Intuitively, the local dimensions of a measure should not be affected by the geometry of the measure on a density zero set of scales. Thus one could expect that tangent distributions should encode all information on dimensions.

Theorem 8 (Hochman (2010+)). If PP is a fractal distribution, then PP almost every measure is exact dimensional. Furthermore, if PP is ergodic, then the value of the dimension is PP almost everywhere constant.

The dimension of a fractal distribution PP is

dimP=dimμdP(μ).\dim P = \int \dim\mu \,\mathrm{d} P(\mu).

It is straightforward to see that the function PdimPP \mapsto \dim P defined on the family of fractal distributions is continuous.

Theorem 9 (Hochman (2010+)). For any Radon measure μ\mu and for μ\mu almost every xx, the pointwise dimensions satisfy

dimloc(μ,x)=sup{dimP:PTD(μ,x) is a fractal distribution},dimloc(μ,x)=inf{dimP:PTD(μ,x) is a fractal distribution}.\begin{align*} \overline{\dim}_{\mathrm{loc}}(\mu,x) &= \sup\{ \dim P : P \in \mathcal{TD}(\mu,x) \text{ is a fractal distribution} \}, \\ \underline{\dim}_{\mathrm{loc}}(\mu,x) &= \inf\{ \dim P : P \in \mathcal{TD}(\mu,x) \text{ is a fractal distribution} \}. \end{align*}

In particular, if μ\mu is a USM generating a fractal distribution PP, then μ\mu is exact dimensional and dimμ=dimP\dim\mu = \dim P.

Conical densities and rectifiability

Let G(d,dk)G(d,d-k) denotes the space of all (dk)(d-k)-dimensional linear subspaces of Rd\mathbb{R}^d and set Sd1={xRd:x=1}S^{d-1} = \{ x \in \mathbb{R}^d : |x|=1 \}. For xRdx \in \mathbb{R}^d, r>0r>0, VG(d,dk)V \in G(d,d-k), θSd1\theta \in S^{d-1}, and 0<α10<\alpha\le 1 define

X(x,r,V,α)={yB(x,r):dist(yx,V)<αyx},H(x,θ,α)={yRd:(yx)θαyx}.\begin{align*} X(x,r,V,\alpha) &= \{ y \in B(x,r) : \mathrm{dist}(y-x,V) < \alpha|y-x| \}, \\ H(x,\theta,\alpha) &= \{ y \in \mathbb{R}^d : (y-x) \cdot \theta \ge \alpha|y-x| \}. \end{align*}

For example, the set X(x,r,V,α)H(x,θ,α)X(x,r,V,\alpha) \setminus H(x,\theta,\alpha) when d=3d=3 and k=1k=1 is as follows:

Conical density results aim to give conditions on a measure (for example, a lower bound on some dimension) which guarantee that the non-symmetric cones X(x,r,V,α)H(x,θ,α)X(x,r,V,\alpha) \setminus H(x,\theta,\alpha) contain a large portion of the mass from the surrounding ball B(x,r)B(x,r) for certain proportion of scales.

Theorem 10 (K-Sahlsten-Shmerkin (2015)). If dNd \in \mathbb{N}, k{1,,d1}k \in \{ 1,\ldots,d-1 \}, k<sdk < s \le d, and 0<α10 < \alpha \le 1, then there exists 0<ε<ε(d,k,α)0 < \varepsilon < \varepsilon(d,k,\alpha) satisfying the following: For every Radon measure μ\mu on Rd\mathbb{R}^d with dimHμs\underline{\dim}_{\mathrm{H}} \mu \ge s it holds that

lim infT1TL1({t[0,T]:infθSd1VG(d,dk)μ(X(x,et,V,α)H(x,θ,α))μ(B(x,et))>ε})skdk\liminf_{T \to \infty} \frac{1}{T} \,\mathcal{L}^1\Bigl(\Bigl\{ t \in [0,T] : \inf_{\genfrac{}{}{0pt}{}{\theta \in S^{d-1}}{V \in G(d,d-k)}} \frac{\mu(X(x,e^{-t},V,\alpha) \setminus H(x,\theta,\alpha))}{ \mu(B(x,e^{-t}))} > \varepsilon \Bigr\}\Bigr) \ge \frac{s-k}{d-k}

at μ\mu almost every xRdx \in \mathbb{R}^d. If the measure μ\mu only satisfies dimpμs\underline{\dim}_{\mathrm{p}}\mu \ge s, then the above inequality holds with lim supT\limsup_{T \to \infty} at μ\mu almost every xRdx \in \mathbb{R}^d.

         

Tangent distributions are well suited to address problems concerning conical densities. The cones in question do not change under magnification and this allows to pass information between the original measure and its tangent distributions. The inequality in Theorem 10 can be restated as

lim infTμx,T(Bε)skdk,\liminf_{T \to \infty} \langle \mu \rangle_{x,T}(\mathcal{B}_\varepsilon) \ge \frac{s-k}{d-k},

where

Bε={νM1:infθSd1VG(d,dk)ν(X(0,1,V,α)H(0,θ,α))>ε}.\mathcal{B}_\varepsilon = \bigl\{ \nu \in \mathcal{M}_1 : \inf_{\genfrac{}{}{0pt}{}{\theta \in S^{d-1}}{V \in G(d,d-k)}} \nu(X(0,1,V,\alpha) \setminus H(0,\theta,\alpha)) > \varepsilon \bigl\}.

This is the main link between the geometric problem we consider and the scenery flow. Recalling Theorem 6, we let μ\mu be the uniformly scaling measure generating

P=skdkδL+(1skdk)δH,P = \frac{s-k}{d-k} \delta_\mathcal{L} + \Bigl( 1-\frac{s-k}{d-k} \Bigr) \delta_\mathcal{H},

where L\mathcal{L} is the normalization of LdB1\mathcal{L}^d|_{B_1} and H\mathcal{H} is the normalization of HkWB1\mathcal{H}^k|_{W \cap B_1} for a fixed WG(d,k)W \in G(d,k). By Theorem 8, the Radon measure μ\mu is of exact dimension ss and it can be shown that the limit limTμx,T(Bε)\lim_{T \to \infty} \langle \mu \rangle_{x,T}(\mathcal{B}_\varepsilon) exists and equals skdk\frac{s-k}{d-k} for all small enough ε>0\varepsilon > 0. Therefore, the inequality in Theorem 10 can be considered sharp.

The proof of Theorem 10 is based on showing that there cannot be “too many” rectifiable tangent measures. This means that, perhaps surprisingly, most of the known conical density results are, in some sense, a manifestation of rectifiability. A set ERdE \subset \mathbb{R}^d is called kk-rectifiable if there are countably many Lipschitz maps fi ⁣:RkRdf_i \colon \mathbb{R}^k \to \mathbb{R}^d so that

Hk(Eifi(Rk))=0.\mathcal{H}^k\Bigl( E \setminus \bigcup_i f_i(\mathbb{R}^k) \Bigr) = 0.

Moreover, we say that a Radon measure ν\nu is kk-rectifiable if νHk\nu \ll \mathcal{H}^k and there exists a kk-rectifiable set ERdE\subset\mathbb{R}^d such that ν(RdE)=0\nu(\mathbb{R}^d\setminus E)=0. A set ERdE \subset \mathbb{R}^d is called strongly kk-rectifiable if there exist countably many Lipschitz maps fi ⁣:RkRdf_i \colon \mathbb{R}^k \to \mathbb{R}^d so that

Eifi(Rk).E \subset \bigcup_i f_i(\mathbb{R}^k).

A Radon measure μ\mu is purely kk-unrectifiable if it gives no mass to kk-rectifiable sets and EE is purely kk-unrectifiable if the restriction HkE\mathcal{H}^k|_E is purely kk-unrectifiable. A strongly kk-rectifiable set is obviously kk-rectifiable. Any set ERdE \subset \mathbb{R}^d with Hk(E)=0\mathcal{H}^k(E) = 0 and dimp(E)>k\dim_{\mathrm{p}}(E)>k is kk-rectifiable but not strongly kk-rectifiable. Furthermore, it follows from analyst’s traveling salesman theorem that any set of upper Minkowski dimension strictly less than 11 is strongly 11-rectifiable.

About the proof

Let us sketch the proof of Theorem 10. For each 0<α10<\alpha\le 1, defining a closed set Aε=M1Bε\mathcal{A}_\varepsilon = \mathcal{M}_1 \setminus \mathcal{B}_\varepsilon, i.e.

Aε={νM1:ν(X(0,1,V,α)H(0,θ,α))ε for some VG(d,dk) and θSd1},\mathcal{A}_\varepsilon = \bigl\{ \nu \in \mathcal{M}_1 : \nu(X(0,1,V,\alpha) \setminus H(0,\theta,\alpha)) \le \varepsilon \text{ for some } V \in G(d,d-k) \text{ and } \theta \in S^{d-1} \bigl\},

the task is to show that there exists a small enough ε>0\varepsilon > 0 such that every Radon measure μ\mu on Rd\mathbb{R}^d with dimHμs\underline{\dim}_{\mathrm{H}} \mu \ge s satisfies lim supTμx,T(Aε)1skdk\limsup_{T \to \infty} \langle \mu \rangle_{x,T}(\mathcal{A}_\varepsilon) \le 1 - \frac{s-k}{d-k} at μ\mu almost every xRdx \in \mathbb{R}^d. The proof of Theorem 10 is basically just the machinery of fractal distributions and the following rectifiability criterion.

Lemma 11. A set ERdE \subset \mathbb{R}^d is strongly kk-rectifiable if for every xEx \in E there are VG(d,dk)V \in G(d,d-k), θSd1\theta \in S^{d-1}, 0<α<1,0 < \alpha < 1, and r>0r > 0 so that EX(x,r,V,α)H(x,θ,α)=E \cap X(x,r,V,\alpha) \setminus H(x,\theta,\alpha) = \emptyset.

Let 0<p<skdk0 < p < \frac{s-k}{d-k}. Suppose to the contrary that there is 0<α10<\alpha\le 1 so that for each small enough ε>0\varepsilon > 0 there exists a Radon measure μ\mu with dimHμs\underline{\dim}_{\mathrm{H}}\mu \ge s such that

lim supTμx,T(Aε)>1p,\limsup_{T \to \infty}\,\langle \mu \rangle_{x,T}(\mathcal{A}_\varepsilon) > 1-p,

on a set EεE_\varepsilon of positive μ\mu measure. Fix δ>0\delta>0 such that p<sδkdkp < \frac{s-\delta-k}{d-k}. Relying on Theorem 3 and Theorem 9, we may assume that all tangent distributions of μ\mu at points xEεx \in E_{\varepsilon} are fractal distributions, and

inf{dimP:PTD(μ,x)}=dimloc(μ,x)>sδ.\inf\{\dim P : P \in \mathcal{TD}(\mu,x)\} = \underline{\dim}_{\mathrm{loc}}(\mu,x) > s-\delta.

Fix xEεx \in E_{\varepsilon}. For each small enough ε>0\varepsilon > 0, as Aε\mathcal{A}_\varepsilon is closed, we find a tangent distribution PεTD(μ,x)P_{\varepsilon} \in \mathcal{TD}(\mu,x) so that Pε(Aε)1pP_{\varepsilon}(\mathcal{A}_\varepsilon) \ge 1-p. Since the sets Aε\mathcal{A}_\varepsilon are nested and closed, we see that

P(A0)=limε0P(Aε)1p,P(\mathcal{A}_0) = \lim_{\varepsilon \downarrow 0} P(\mathcal{A}_\varepsilon) \ge 1-p,

where PP is a weak^* limit of a sequence formed by PεP_\varepsilon as ε0\varepsilon \downarrow 0. Furthermore, since the collection of all fractal distributions is closed by Theorem 4 and the dimension is continuous, the limit distribution PP is a fractal distribution with

dimPsδ.\dim P \ge s-\delta.

Recalling Theorem 2, let

P=PωdP(ω)P = \int P_\omega \,\mathrm{d} P(\omega)

be the ergodic decomposition of PP. By the invariance of A0\mathcal{A}_0, we have Pω(A0){0,1}P_\omega(\mathcal{A}_0) \in \{0,1\} for PP almost all ω\omega. If Pω(A0)=0P_\omega(\mathcal{A}_0) = 0, we use the trivial estimate

dimPωd.\dim P_\omega \le d.

If Pω(A0)=1P_\omega(\mathcal{A}_0) = 1, then, using the quasi-Palm property, for PωP_\omega almost every ν\nu and for ν\nu almost every zz the normalized translation νz,tz\nu_{z,t_z} is an element of A0\mathcal{A}_0 for some tz>0t_z > 0 with B(z,etz)B1B(z,e^{-t_z}) \subset B_1. For each such ν\nu let

E={zB1:νz,tzA0}E = \{ z \in B_1 : \nu_{z,t_z} \in \mathcal{A}_0 \}

be this set of full ν\nu measure. Thus for every zEz \in E there are VG(d,dk)V \in G(d,d-k) and θSd1\theta \in S^{d-1} with

EX(z,etz,V,α)H(z,θ,α)=.E \cap X(z,e^{-t_z},V,\alpha) \setminus H(z,\theta,\alpha) = \emptyset.

Lemma 11 implies that EE is strongly kk-rectifiable. In particular, dimνk\dim \nu \le k, which yields dimPωk\dim P_\omega \le k. Since P({ω:Pω(A0)=1})=P(A0)1pP(\{ \omega : P_\omega(\mathcal{A}_0) = 1 \}) = P(\mathcal{A}_0) \ge 1-p we estimate

sδdimP=dimPωdP(ω)P(A0)k+(1P(A0))d(1p)k+pds-\delta \leq \dim P = \int \dim P_\omega \,\mathrm{d} P(\omega) \leq P(\mathcal{A}_0)k + (1-P(\mathcal{A}_0))d \le (1-p)k + pd

which gives psδkdkp \ge \frac{s-\delta-k}{d-k}. But this contradicts the choice of δ\delta.

Average unrectifiability

Pure unrectifiability is also a condition which guarantees that the measure is scattered in many directions. We introduce average unrectifiability and show that it implies a conical density result. Under certain assumption, we also show the converse. Given a proportion 0p<10 \leq p < 1, we say that a Radon measure μ\mu is pp-average kk-unrectifiable if we have

P({νM1:sptν is not strongly k-rectifiable})>pP(\{\nu \in \mathcal{M}_1 : \mathrm{spt}\nu \text{ is not strongly } k \text{-rectifiable}\}) > p

for every PTD(μ,x)P \in \mathcal{TD}(\mu,x) at μ\mu almost every xx.

Theorem 12 (K-Sahlsten-Shmerkin (2015)). Suppose that dNd \in \mathbb{N}, k{1,,d1}k \in \{ 1,\ldots,d-1 \}, and 0p<10 \le p < 1. If μ\mu is pp-average kk-unrectifiable, then for every 0<α10 < \alpha \le 1 there exists 0<ε<10 < \varepsilon < 1 so that

lim infT1Tμx,T(Bε)>p\liminf_{T \to \infty} \frac{1}{T} \langle \mu \rangle_{x,T}(\mathcal{B}_\varepsilon) > p

at μ\mu almost every xRdx \in \mathbb{R}^d.

Since, by Theorem 10, the critical dimension for conical densities around kk-planes is precisely kk, it is perhaps natural to try to prove the converse for measures of this dimension.

Theorem 13 (K-Sahlsten-Shmerkin (2015)). Suppose that dNd \in \mathbb{N}, k{1,,d1}k \in \{ 1,\ldots,d-1 \}, 0p<10 \le p < 1, 0<α,ε<10<\alpha,\varepsilon<1, and a Radon measure μ\mu satisfies

0<lim infr0μ(B(x,r))rklim supr0μ(B(x,r))rk<0 < \liminf_{r \downarrow 0} \frac{\mu(B(x,r))}{r^k} \leq \limsup_{r \downarrow 0} \frac{\mu(B(x,r))}{r^k} < \infty

and

lim infT1Tμx,T(Bε)>p\liminf_{T \to \infty} \frac{1}{T} \langle \mu \rangle_{x,T}(\mathcal{B}_\varepsilon) > p

at μ\mu almost all xRdx \in \mathbb{R}^d. Then for μ\mu is pp-average kk-unrectifiable.