# Visible parts and slices of Ahlfors regular sets

Damian Dąbrowski  Department of Mathematics and Statistics
University of Jyväskylä, P.O. Box 35 (MaD)
FI-40014 University of Jyväskylä
Finland<EMAIL_ADDRESS>
###### Abstract.

We show that for any compact set $E\subset\mathbb{R}^{d}$ the visible part of
$E$ has Hausdorff dimension at most $d-1/6$ for almost every direction. This
improves recent estimates of Orponen and Matheus.

If $E$ is $s$-Ahlfors regular, where $s>d-1$, we prove a much better estimate.
In that case we have for almost every direction $\theta$

$\operatorname{\dim_{\mathrm{H}}}(\operatorname{Vis}_{\theta}(E))\leq
s-\alpha(s-d+1),$

where $\alpha>0.183$ is absolute. The estimate is new even for self-similar
sets satisfying the open set condition. Along the way, we prove a refinement
of the Marstrand’s slicing theorem for Ahlfors regular sets.

###### Key words and phrases:

visibility conjecture, Hausdorff dimension, Marstrand’s slicing theorem

###### 2020 Mathematics Subject Classification:

28A80 (primary) 28A78 (secondary)

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Slicing Ahlfors regular sets
  4. 4 Decomposition of the visible part
  5. 5 Heavy part
  6. 6 Bad part
  7. 7 Good part
  8. 8 General compact sets

##  1\. Introduction

###  1.1. Visibility conjecture

Given $\theta\in\mathbb{S}^{d-1}$ let $\ell_{\theta}=\\{t\theta\
\mathrel{\mathop{\ordinarycolon}}\ t\geq 0\\}\subset\mathbb{R}^{d}$ be the
closed half-line spanned by $\theta$. The _visible part_ of a compact set
$E\subset\mathbb{R}^{d}$ in direction $\theta$ is defined as

$\operatorname{Vis}_{\theta}(E)=\\{x\in E\ \mathrel{\mathop{\ordinarycolon}}\
(x+\ell_{\theta})\cap E=\\{x\\}\\}.$

Let
$\pi_{\theta}\mathrel{\mathop{\ordinarycolon}}\mathbb{R}^{d}\to\theta^{\perp}$
be the orthogonal projection to the $(d-1)$-dimensional plane
$\theta^{\perp}$. Since
$\pi_{\theta}(\operatorname{Vis}_{\theta}(E))=\pi_{\theta}(E)$, it follows
from the Marstrand-Mattila projection theorem (see e.g. [Mat15, Theorem 5.8])
that for $\mathcal{H}^{d-1}$-a.e. $\theta\in\mathbb{S}^{d-1}$

$\operatorname{\dim_{\mathrm{H}}}(\operatorname{Vis}_{\theta}(E))\geq\min\\{\operatorname{\dim_{\mathrm{H}}}(E),\,d-1\\},$
(1.1)

where $\operatorname{\dim_{\mathrm{H}}}$ stands for the Hausdorff dimension.
Observe that we also have the trivial upper bound

$\operatorname{\dim_{\mathrm{H}}}(\operatorname{Vis}_{\theta}(E))\leq\operatorname{\dim_{\mathrm{H}}}(E)$
(1.2)

simply because $\operatorname{Vis}_{\theta}(E)\subset E$. The two estimates
together imply that for sets satisfying
$\operatorname{\dim_{\mathrm{H}}}(E)\leq d-1$ the inequality (1.1) is in fact
an equality. The _visibility conjecture_ asserts that the same holds for all
compact sets $E$.

######  Conjecture.

If $E\subset\mathbb{R}^{d}$ is compact and
$\operatorname{\dim_{\mathrm{H}}}(E)>d-1$, then for $\mathcal{H}^{d-1}$-a.e.
$\theta\in\mathbb{S}^{d-1}$

$\operatorname{\dim_{\mathrm{H}}}(\operatorname{Vis}_{\theta}(E))=d-1.$ (1.3)

To the best of our knowledge the conjecture first appeared in [JJMO03];
another early mention is [Mat04, Problem 11].

The visibility conjecture has been confirmed for several special classes of
sets. It holds for quasicircles, graphs of continuous functions [JJMO03],
fractal percolation [AJJ+12], and for self-similar and self-affine sets
satisfying various additional hypotheses [JJMO03, FF13, Ros21, JJSW22].

Some progress has also been made towards proving (1.3) for general compact
sets. A special case of Theorem 1.1 from [JJN04] gives that if
$E\subset\mathbb{R}^{d}$ satisfies $0<\mathcal{H}^{s}(E)<\infty$, then
$\mathcal{H}^{s}(\operatorname{Vis}_{\theta}(E))=0$ for a.e.
$\theta\in\mathbb{S}^{d-1}$. In a recent breakthrough, Orponen showed that for
any compact $E\subset\mathbb{R}^{d}$ we have for a.e.
$\theta\in\mathbb{S}^{d-1}$

$\operatorname{\dim_{\mathrm{H}}}(\operatorname{Vis}_{\theta}(E))\leq
d-\frac{1}{50d},$ (1.4)

so that at least for sets with dimension close enough to $d$ we can beat the
trivial bound (1.2). Orponen’s proof was optimised by Matheus [Mat21a], but
due to typos in equations [Mat21a, (2.9) and (3.17)] the estimates obtained
are weaker than stated.

Quite amazingly, in general it is still not known whether for sets satisfying
$\operatorname{\dim_{\mathrm{H}}}(E)>d-1$ we have
$\operatorname{\dim_{\mathrm{H}}}(\operatorname{Vis}_{\theta}(E))<\operatorname{\dim_{\mathrm{H}}}(E)$
for a.e. $\theta\in\mathbb{S}^{d-1}$. Until now, this was open even for self-
similar sets satisfying the open set condition. If we assume additionally that
the rotation group of the associated IFS is finite, and that the projection
$\pi_{\theta}(E)$ has non-empty interior, then this was shown in [JJSW22,
Theorem 2.11]. Some estimates for self-similar sets were also shown in
[Mat21a], but they were only valid if $\operatorname{\dim_{\mathrm{H}}}(E)$
was large enough, and due to the typo in [Mat21a, (3.17)] it is not clear to
us what the estimate obtained for
$\operatorname{\dim_{\mathrm{H}}}(\operatorname{Vis}_{\theta}(E))$ is.

In the first result of this article we improve on the trivial bound (1.2) for
all Ahlfors regular sets with $\operatorname{\dim_{\mathrm{H}}}(E)>d-1$.
Recall that a compact set $E\subset\mathbb{R}^{d}$ is $s$-Ahlfors regular,
where $0<s\leq d$, if there exists a constant $C\geq 1$ such that

$C^{-1}r^{s}\leq\mathcal{H}^{s}(E\cap B(x,r))\leq Cr^{s}\quad\text{for all
$x\in E,\,r>0$.}$

######  Theorem 1.1.

Let $d-1<s\leq d$. Suppose that $E\subset\mathbb{R}^{d}$ is a compact
$s$-Ahlfors regular set. Then, for a.e. $\theta\in\mathbb{S}^{d-1}$

$\operatorname{\dim_{\mathrm{H}}}(\operatorname{Vis}_{\theta}(E))\leq
s-\alpha(s-d+1)=(1-\alpha)s+\alpha(d-1),$ (1.5)

where $\alpha=1-\sqrt{6}/3>0.1835.$

In particular, the result holds for self-similar sets satisfying weak
separation condition, see [FHOR15, Theorem 2.1]. Note that proving (1.5) with
$\alpha=1$ would establish the visibility conjecture for Ahlfors-regular sets.

In [Orp22, Remark 1.3] Orponen remarked that while the constant $1/50$ in
(1.4) could be improved by optimising the argument (this was done in
[Mat21a]), “it seems likely that more ideas will be needed to get an upper
bound of the form $n-c$ for some absolute $c>0$.” In our second main result we
obtain such a bound.

######  Theorem 1.2.

Suppose that $E\subset\mathbb{R}^{d}$ is compact. Then, for a.e.
$\theta\in\mathbb{S}^{d-1}$

$\operatorname{\dim_{\mathrm{H}}}(\operatorname{Vis}_{\theta}(E))\leq
d-\frac{1}{6}.$ (1.6)

This improves the best known upper bound for
$\operatorname{\dim_{\mathrm{H}}}(\operatorname{Vis}_{\theta}(E))$ for general
compact sets.

###  1.2. Slicing Ahlfors regular sets

When proving Theorem 1.1 we needed an upper bound for the dimension of _line
slices_ of Ahlfors regular sets, that is, sets of the form $F\cap\ell$, where
$\ell$ is a line. We prove a more general result, valid for slices of
arbitrary dimension.

Let $1\leq n\leq d-1$ be an integer. The celebrated Marstrand’s slicing
theorem [Mar54, Mat75] states that for a set $F\subset\mathbb{R}^{d}$ with
$0<\mathcal{H}^{s}(F)<\infty$ and $n<s\leq d$ we have for $\gamma_{d,n}$-a.e.
$L\in\mathbb{\mathcal{G}}(d,n)$

$\operatorname{\dim_{\mathrm{H}}}(F\cap(x+L^{\perp}))=s-n\quad\text{for
$\mathcal{H}^{s}$-a.e. $x\in F$}.$ (1.7)

Here $\mathcal{G}(d,n)$ denotes the Grassmannian of $n$-dimensional planes in
$\mathbb{R}^{d}$, and $\gamma_{d,n}$ is the Haar measure on
$\mathcal{G}(d,n)$. See [Mat95, Theorem 10.11] or [Mat15, Theorem 6.9] for
textbook references.

Refinements and alternative proofs of Marstrand’s slicing theorem have been
found in [Mat81, Orp14, MO16], see also [Mat15, Chapter 6] and recent surveys
[Mat21b, Mat23]. The properties of slices of certain special classes of
fractals have also been extensively studied, see e.g. [Wu19, Shm19, Alg20,
BKY21, ABK23].

In [Mat15, p. 97] Mattila asked whether it is possible to get a dimension
upper bound on the set

$F_{u,L}\coloneqq\\{x\in F\ \mathrel{\mathop{\ordinarycolon}}\
\operatorname{\dim_{\mathrm{H}}}(F\cap(x+L^{\perp})>u\\},$

where $0<\mathcal{H}^{s}(F)<\infty$, $n<s\leq d$, and $s-n\leq u<d-n$. Note
that (1.7) gives only $\mathcal{H}^{s}(F_{u,L})=0$ for a.e.
$L\in\mathcal{G}(d,n)$. We prove such estimate for the endpoint $u=s-n$ in the
case that $F$ is Ahlfors regular. In this case, we are also able with no extra
effort to replace in the definition of $F_{u,L}$ the Hausdorff dimension by
the upper box-counting dimension.

######  Theorem 1.3.

Let $n<s\leq d$. Suppose that $F\subset\mathbb{R}^{d}$ is a compact
$s$-Ahlfors regular set. Then, for $\gamma_{d,n}$-a.e. $L\in\mathcal{G}(d,n)$

$\operatorname{\dim_{\mathrm{H}}}(\\{x\in F\
\mathrel{\mathop{\ordinarycolon}}\ \overline{\dim}_{\mathrm{box}}(F\cap
V_{x,L})>s-n\\})\leq n.$ (1.8)

For a more quantitative version of Theorem 1.3, see Proposition 3.1.

######  Remark 1.4.

In fact, we prove that $\eqref{eq:slicing}$ holds for every
$L\in\mathcal{G}(d,n)$ such that the push-forward of $\mathcal{H}^{s}|_{F}$
under the orthogonal projection $\pi_{L}$ belongs to the Sobolev space
$H^{\sigma}(\mathbb{R}^{n})$ for all $0<\sigma<(s-n)/2$. This is known to be
true for $\gamma_{d,n}$-a.e. $L\in\mathcal{G}(d,n)$ (see Lemma 2.4), and for a
general set $F\subset\mathbb{R}^{d}$ with $0<\mathcal{H}^{s}(F)<\infty$ this
is likely to be sharp, in the sense that the set of exceptional planes
$\mathcal{E}\subset\mathcal{G}(d,n)$ where the Sobolev bound fails may satisfy
$\operatorname{\dim_{\mathrm{H}}}(\mathcal{E})=\operatorname{\dim_{\mathrm{H}}}(\mathcal{G}(d,n))$.
However, it seems plausible that for Ahlfors regular sets $F$ the set of
exceptional planes $\mathcal{E}$ is much smaller. Hence, proving any non-
trivial upper bounds on $\operatorname{\dim_{\mathrm{H}}}(\mathcal{E})$ for
Ahlfors regular sets would immediately give an estimate on the dimension of
exceptional planes where $\eqref{eq:slicing}$ fails.

######  Remark 1.5.

The estimate (1.8) is only interesting if $s\leq 2n$, because for $s>2n$ it is
easy to show that for $\gamma_{d,n}$-a.e. $L\in\mathcal{G}(d,n)$

$\\{x\in F\ \mathrel{\mathop{\ordinarycolon}}\
\overline{\dim}_{\mathrm{box}}(F\cap V_{x,L})>s-n\\}=\varnothing,$ (1.9)

see the argument above (3.11). (1.9) also holds trivially for $s=d$, simply
because $\operatorname{\dim_{\mathrm{H}}}(V_{x,L})=d-n$.

######  Question 1.6.

It is not clear to us how sharp (1.8) is. Comparing (1.8) and (1.9), it is
tempting to ask whether for $n<s\leq\min(2n,d)$ and sets with
$0<\mathcal{H}^{s}(F)<\infty$ we have

$\operatorname{\dim_{\mathrm{H}}}(\\{x\in F\
\mathrel{\mathop{\ordinarycolon}}\ \operatorname{\dim_{\mathrm{H}}}(F\cap
V_{x,L})>s-n\\})\leq 2n-s$

for a.e. $L\in\mathcal{G}(d,n)$.

Observe that if this estimate was true, then for $s>3n/2$ it would
automatically improve to $\\{x\in F\ \mathrel{\mathop{\ordinarycolon}}\
\operatorname{\dim_{\mathrm{H}}}(F\cap V_{x,L})>s-n\\}=\varnothing$ for a.e.
$L\in\mathcal{G}(d,n)$, simply because $s-n>2n-s$ in this regime.

###  1.3. About the proofs

The proofs of Theorem 1.1 and Theorem 1.2 follow the general strategy invented
in [Orp22], which we briefly describe below.

Given a compact set $E\subset[0,1]^{d}$ and a fixed direction
$\theta\in\mathbb{S}^{d-1}$, we seek an estimate for
$\mathcal{H}^{d-\tau}_{\infty}(\operatorname{Vis}_{\theta}(E))$ for some
$\tau>0$. Consider the lines $\mathcal{L}_{\theta}$ parallel to $\theta$ that
cover $[0,1]^{d}$. The lines are divided into two classes, the good lines and
the bad lines. Loosely speaking, a line $\ell\in\mathcal{L}_{\theta}$ is good
if the slice $E\cap\ell$ is quite similar to the $\delta$-slice
$E\cap\ell(\delta)$, where $\ell(\delta)$ is the $\delta$-neighbourhood of
$\ell$. Otherwise, $\ell$ is bad. Let $L_{B}$ be the union of bad lines, and
$L_{G}$ the union of good lines. We estimate
$\mathcal{H}^{d-\tau}_{\infty}(\operatorname{Vis}_{\theta}(E)\cap L_{B})$ and
$\mathcal{H}^{d-\tau}_{\infty}(\operatorname{Vis}_{\theta}(E)\cap L_{G})$
separately.

We have a poor understanding of how $\operatorname{Vis}_{\theta}(E)\cap L_{B}$
looks, but luckily we can show that there are very few bad lines, in the sense
that $\pi_{\theta}(L_{B})$ has small dimension. This step uses crucially the
Sobolev estimate for the projections of high-dimensional Frostman measures,
see Lemma 2.4. Then, we can estimate crudely

$\mathcal{H}^{d-\tau}_{\infty}(\operatorname{Vis}_{\theta}(E)\cap
L_{B})\leq\mathcal{H}^{d-\tau-1}_{\infty}(\pi_{\theta}(L_{B}))\lesssim\delta^{\varepsilon}.$
(1.10)

On the other hand, the good part $\operatorname{Vis}_{\theta}(E)\cap L_{G}$
has a very nice structure, so that for each $\delta$-tube $T$ parallel to
$\theta$ we have

$N(T\cap\operatorname{Vis}_{\theta}(E)\cap
L_{G},\delta)\lesssim\delta^{\tau-1+\varepsilon},$ (1.11)

where $N(\cdot,\delta)$ stands for $\delta$-covering number, and then

$\mathcal{H}^{d-\tau}_{\infty}(\operatorname{Vis}_{\theta}(E)\cap
L_{G})\lesssim\sum_{T}N(T\cap\operatorname{Vis}_{\theta}(E)\cap
L_{G},\delta)\cdot\delta^{d-\tau}\lesssim\delta^{1-d}\cdot\delta^{\tau-1+\varepsilon}\cdot\delta^{d-\tau}=\delta^{\varepsilon}.$

This is the outline of the proof in [Orp22]. In this paper we make two
improvements. Firstly, in [Orp22] one begins by discarding exceptional
directions where the Sobolev norm of projections is large (this is important
for the estimate of $\mathcal{H}^{d-\tau-1}_{\infty}(\pi_{\theta}(L_{B}))$).
In the end, this forces $\tau\lesssim 1/d$. In the current paper, we do not
discard exceptional directions. Instead, all our estimates depend on the
Sobolev norm of the projections, and only at the end we average over all the
directions. This removes some annoying terms from the estimates, and allows us
to obtain Theorem 1.2, where $\tau$ is independent of $d$.

The second improvement consists of using the slicing estimate from Theorem 1.3
(or rather Proposition 3.1) in the case $n=d-1$, which corresponds to slicing
with lines. In [Orp22] it is hard to do better than (1.10) or (1.11) because a
priori each slice might be of dimension 1. Using Theorem 1.3, we know that for
an $s$-Ahlfors regular set $E$ most slices will have dimension at most
$s-d+1$, which allows us to significantly improve (1.10) and (1.11). This is
how we get Theorem 1.1.

Finally, the proof of Theorem 1.3 relies on the Sobolev regularity of
projections (Lemma 2.4) and the trace formulas for Sobolev functions (Theorem
2.1). It involves proving a weak-type estimate for the Hardy-Littlewood
maximal function which is roughly of the form
$\mathcal{M}\mathrel{\mathop{\ordinarycolon}}H^{\sigma}(\mathbb{R}^{n})\to
L^{2,\infty}(\mathcal{H}^{n-2\sigma}_{\infty})$ for $0<\sigma<n/2$, see Lemma
3.2.

### Acknowledgments

I am grateful to Tuomas Orponen for many helpful discussions, and to Esa
Järvenpää for reading a draft of this paper and giving useful comments.

The project was supported by the Academy of Finland grant Quantitative
rectifiability and harmonic measure beyond the Ahlfors-David-regular setting,
grant No. 347123.

##  2\. Preliminaries

###  2.1. Notation

Inequalities of the form $f\leq Cg$, where $C$ is some constant, will be
abbreviated as $f\lesssim g$. If the constant $C$ depends on an additional
parameter $\varepsilon$, we will write $f\lesssim_{\varepsilon}g$. We will
ignore dependence on dimension $d$ and on Ahlfors-regularity constants. If we
have $g\lesssim f\lesssim g$, we will write $f\sim g$.

Given $x\in\mathbb{R}^{d},r>0$ we will denote the open ball centered at $x$ of
radius $r$ by $B(x,r)$. If $x\in\mathbb{R}^{d-1}$, then $B(x,r)$ will stand
for the $(d-1)$-dimensional open ball in $\mathbb{R}^{d-1}$. If $B$ is a ball,
then $r(B)$ will denote its radius.

If $A\subset\mathbb{R}^{d}$ and $\delta>0$, then $N(A,\delta)$ denotes the
$\delta$-covering number of $A$.

Given an affine line $\ell$ and $\delta>0$, $\ell(\delta)$ will denote the
$\delta$-neighbourhood of $\ell$.

Let $\mathbb{D}$ be the family of standard dyadic cubes on $\mathbb{R}^{d}$.
Given a dyadic cube $Q$ we will denote its sidelength by $\ell(Q)$.

$\mathcal{G}(d,n)$ stands for the Grassmannian of $n$-dimensional planes in
$\mathbb{R}^{d}$. Given $L\in\mathcal{G}(d,n)$,
$\pi_{L}\mathrel{\mathop{\ordinarycolon}}\mathbb{R}^{d}\to
L\simeq\mathbb{R}^{n}$ denotes the orthogonal projection to $L$. Given
$\theta\in\mathbb{S}^{d-1}$, we set
$\pi_{\theta}\coloneqq\pi_{\theta^{\perp}}$, so that
$\pi_{\theta}\mathrel{\mathop{\ordinarycolon}}\mathbb{R}^{d}\to\mathbb{R}^{d-1}$.

###  2.2. Sobolev spaces

For $\sigma\in\mathbb{R}$ the fractional Sobolev space
$H^{\sigma}(\mathbb{R}^{d})\subset\mathcal{S}^{\prime}(\mathbb{R}^{d})$ is the
completion of the set of Schwartz functions $f\in\mathcal{S}(\mathbb{R}^{d})$
such that

$\|f\|_{H^{\sigma}(\mathbb{R}^{d})}^{2}\coloneqq\int|\widehat{f}(\xi)|^{2}(1+|\xi|^{2})^{\sigma}\
d\xi<\infty.$

The quantity above is the inhomogeneous Sobolev norm. Its homogeneous
counterpart is

$\|f\|_{\dot{H}^{\sigma}(\mathbb{R}^{d})}^{2}\coloneqq\int|\widehat{f}(\xi)|^{2}|\xi|^{2\sigma}\
d\xi.$

Observe that if $\nu$ is a finite Borel measure on $\mathbb{R}^{d}$, and
$\sigma\geq 0$, then we can estimate the inhomogeneous norm using the
homogeneous one:

$\|\nu\|_{H^{\sigma}(\mathbb{R}^{d})}^{2}=\int|\widehat{\nu}(\xi)|^{2}(1+|\xi|^{2})^{\sigma}\
d\xi\\\ \lesssim\int_{B(0,1)}|\widehat{\nu}(\xi)|^{2}\
d\xi+\int_{B(0,1)^{c}}|\widehat{\nu}(\xi)|^{2}|\xi|^{2\sigma}\ d\xi\\\
\lesssim\|\widehat{\nu}(\xi)\|_{L^{\infty}}^{2}+\|\nu\|_{\dot{H}^{\sigma}(\mathbb{R}^{d})}^{2}\leq\nu(\mathbb{R}^{d})^{2}+\|\nu\|_{\dot{H}^{\sigma}(\mathbb{R}^{d})}^{2}.$
(2.1)

Another tool we will use in our proof is the theory of traces of Sobolev
functions with respect to fractal measures. Results of this type can be found
e.g. in the classical monographs [Maz11, Chapter 11], [AH96, Chapter 7], or
[Tri06, Chapter 7]. The statement we found particularly convenient was Theorem
7.16 from [Tri06], and in our setting it gives the following.

######  Theorem 2.1.

Let $0<t<d$, $\varepsilon>0$, and $\sigma\geq(d-t+\varepsilon)/2$. Suppose
that $f\in\mathcal{S}(\mathbb{R}^{d})$, and that $\eta$ is a Radon probability
measure on $\mathbb{R}^{d}$ satisfying $\eta(B(x,r))\leq C_{0}\,r^{t}$. Then,

$\|f\|_{L^{2}(\eta)}\lesssim_{\varepsilon}(C_{0})^{1/2}\,\|f\|_{H^{\sigma}(\mathbb{R}^{d})}.$

###  2.3. Frostman measures

We are going to use two versions of the classical Frostman’s lemma. The first
one is due to Orponen [Orp22, Lemma A.1], and the difference compared to the
usual Frostman’s lemma is the lower bound on $\nu(\overline{Q})$.

######  Lemma 2.2.

If $K\subset[0,1)^{d}$ is compact, $0<t\leq d$, then there exists a Radon
measure $\nu$ with $\operatorname{supp}\nu\subset K$, satisfying

$\nu(B(x,r))\lesssim r^{t}\quad\text{for $x\in\mathbb{R}^{d},r>0$}$

and for all dyadic cubes $Q\subset[0,1)^{d}$

$\nu(\overline{Q})\gtrsim\min(\mathcal{H}^{t}_{\infty}(Q\cap
K),\mathcal{H}^{d}(Q)).$

We remark that the Frostman measure $\nu$ above satisfies
$\nu(K)\sim\mathcal{H}^{t}_{\infty}(K)$, and in particular if
$t>\operatorname{\dim_{\mathrm{H}}}(K)$, then $\nu\equiv 0$.

The second version of Frostman’s lemma we will use concerns unions of dyadic
cubes, and it asserts that at small scales the Frostman measure associated to
such sets is just normalized Lebesgue measure. This follows from the usual
construction of the Frostman measure using dyadic cubes, as in [Mat95, Theorem
8.8], and we omit the proof.

######  Lemma 2.3.

Let $\mathcal{K}\subset\mathbb{D}$ be a collection of disjoint dyadic cubes
with $K\coloneqq\bigcup_{Q\in\mathcal{K}}Q\subset[0,1]^{d}$. For any $0<t\leq
d$ there exists a Radon measure $\nu$ with
$\operatorname{supp}\nu\subset\overline{K}$, satisfying
$\nu(K)\sim\mathcal{H}^{t}_{\infty}(K)$,

$\nu(B(x,r))\lesssim r^{t}\quad\text{for $x\in\mathbb{R}^{d},r>0$},$

and for all $Q\in\mathcal{K}$

$\nu|_{Q}=\frac{\nu(Q)}{\ell(Q)^{d}}\mathcal{H}^{d}|_{Q}.$ (2.2)

Let $\nu$ be a compactly supported Radon measure on $\mathbb{R}^{d}$. Given
$L\in\mathcal{G}(d,n)$ we will write $\nu_{L}$ to denote the push-forward of
$\nu$ under the orthogonal projection $\pi_{L}$. Whenever
$\nu_{L}\ll\mathcal{H}^{n},$ which will always be the case in this paper, the
expression “$\nu_{L}$” will denote both the measure and its density with
respect to $\mathcal{H}^{n}$. Similarly, for $\theta\in\mathbb{S}^{d-1}$ we
will write $\nu_{\theta}$ to denote the push-forward of $\nu$ under the
orthogonal projection
$\pi_{\theta}\mathrel{\mathop{\ordinarycolon}}\mathbb{R}^{d}\to\mathbb{R}^{d-1}.$

The following estimate is well-known (see e.g. [Mat15, Theorem 5.10]), so we
only sketch the proof.

######  Lemma 2.4.

Suppose that $\nu$ is Radon measure on $\mathbb{R}^{d}$ with
$\operatorname{supp}\nu\subset[0,1]^{d}$, and that it satisfies
$\nu(B(x,r))\leq C_{0}\,r^{t}$ for some $n<t\leq d$. Let $0<\varepsilon<t-n$,
and $\sigma=(t-n-\varepsilon)/2$. Then,

$\int_{\mathcal{G}(d,n)}\|\nu_{L}\|_{H^{\sigma}(\mathbb{R}^{n})}^{2}\,d\gamma_{d,n}(L)\lesssim_{\varepsilon}C_{0}\,\nu(\mathbb{R}^{d}).$

###### Proof.

Recall that given $0<s<d$, the Riesz $s$-energy of a Radon measure $\mu$ on
$\mathbb{R}^{d}$ is defined as

$I_{s}(\mu)\coloneqq\iint\frac{1}{|x-y|^{s}}\ d\mu(x)d\mu(y).$

If $\mu$ is compactly supported, then we also have the following well-known
identity relating Riesz energy with the homogeneous Sobolev norm: for $0<s<d$

$I_{s}(\mu)=C_{d,s}\int_{\mathbb{R}^{d}}|\widehat{\mu}(\xi)|^{2}|\xi|^{s-d}\,d\xi,$
(2.3)

see [Mat15, Theorem 3.10] for the proof.

A simple computation shows that the Frostman condition $\nu(B(x,r))\leq
C_{0}\,r^{t}$ implies

$I_{t-\varepsilon}(\nu)\lesssim_{\varepsilon}C_{0}\,\nu(\mathbb{R}^{d}).$
(2.4)

Using the fact that $\widehat{\nu_{L}}(\xi)=\widehat{\nu}(\xi)$ for $\xi\in
L$, another standard computation (see [Mat15, Theorem 5.10] for details) gives

$\int_{\mathcal{G}(d,n)}\|\nu_{L}\|_{\dot{H}^{\sigma}(\mathbb{R}^{n})}^{2}\,d\theta=\int_{\mathcal{G}(d,n)}\int_{L}|\widehat{\nu_{L}}(\xi)|^{2}|\xi|^{t-\varepsilon-n}\
d\mathcal{H}^{n}(\xi)\,d\gamma_{d,n}(L)\\\
\sim\int_{\mathbb{R}^{d}}|\widehat{\nu}(\xi)|^{2}|\xi|^{t-\varepsilon-d}\
d\xi\sim
I_{t-\varepsilon}(\nu)\lesssim_{\varepsilon}C_{0}\,\nu(\mathbb{R}^{d}).$

Together with (2.1), this gives the desired estimate. ∎

##  3\. Slicing Ahlfors regular sets

In this section we prove Theorem 1.3. We first prove the following more
quantitative result, which will be crucial later in the proof of Theorem 1.1.

Given a locally integrable function
$f\mathrel{\mathop{\ordinarycolon}}\mathbb{R}^{d}\to\mathbb{R}$, the centered
Hardy-Littlewood maximal function of $f$ is defined as

$\mathcal{M}f(x)=\sup_{r>0}\frac{1}{r^{d}}\int_{B(x,r)}|f(y)|\,dy.$

Recall that if $\nu$ is a Radon measure on $\mathbb{R}^{d}$ and
$L\in\mathcal{G}(d,n)$, then $\nu_{L}$ is the pushforward of $\nu$ under the
orthogonal projection
$\pi_{L}\mathrel{\mathop{\ordinarycolon}}\mathbb{R}^{d}\to\mathbb{R}^{n}$.

######  Proposition 3.1.

Let $n<s\leq\min(2n,d)$ and $0<\varepsilon<(s-n)/2$. Let $\nu$ be an
$s$-Ahlfors regular measure on $\mathbb{R}^{d}$ with
$F\coloneqq\operatorname{supp}\nu\subset B(0,1)$. Let $M>3^{n}\nu(F)$,
$L\in\mathcal{G}(d,n)$, and

$F_{M}=\bigg{\\{}x\in F\ \mathrel{\mathop{\ordinarycolon}}\
\mathcal{M}\nu_{L}(\pi_{L}(x))\geq M\bigg{\\}}.$

Then, for $\sigma=(s-n-\varepsilon)/2$ we have

$\mathcal{H}^{n+2\varepsilon}_{\infty}(F_{M})\lesssim_{\varepsilon}M^{-1}\|\nu_{L}\|^{2}_{H^{\sigma}(\mathbb{R}^{n})}.$

Most of this section is dedicated to proving this result. Fix $\nu$ and $F$ as
above, and let

$H^{\prime}=\\{x\in\mathbb{R}^{n}\ \mathrel{\mathop{\ordinarycolon}}\
\mathcal{M}\nu_{L}(x)\geq M\\},$

so that $H^{\prime}=\pi_{L}(F_{M})$. We would like to get a bound of the form

$\mathcal{H}^{2n-s+2\varepsilon}_{\infty}(H^{\prime})\lesssim_{\varepsilon}M^{-2}\|\nu_{L}\|^{2}_{H^{\sigma}(\mathbb{R}^{n})}.$

We first establish a dyadic variant of this estimate.

Let $\mathcal{B}\subset\mathbb{D}_{\mathbb{R}^{n}}$ be the family of maximal
dyadic cubes satisfying

$\frac{\nu_{L}(Q)}{\ell(Q)^{n}}\geq 3^{-n}M.$ (3.1)

Since $3^{-n}M>\nu(F)$ and $\operatorname{supp}\nu_{L}\subset[0,1)^{n}$, we
get that all cubes from $\mathcal{B}$ are contained in $[0,1)^{n}$. Let
$H=\bigcup_{Q\in\mathcal{B}}Q$.

Note that for any $Q\in\mathbb{D}_{\mathbb{R}^{n}}$ which is not contained in
$H$ we have

$\frac{\nu_{L}(Q)}{\ell(Q)^{n}}<3^{-n}M.$ (3.2)

######  Lemma 3.2.

We have

$\mathcal{H}^{2n-s+2\varepsilon}_{\infty}(H)\lesssim_{\varepsilon}M^{-2}\|\nu_{L}\|^{2}_{H^{\sigma}(\mathbb{R}^{n})}.$
(3.3)

###### Proof.

Assume that $\mathcal{H}^{2n-s+2\varepsilon}_{\infty}(H)>0$, since otherwise
there is nothing to prove. Let $\eta$ be the Radon measure on $H$ given by
Frostman’s lemma Lemma 2.3, normalized so that $\eta(\mathbb{R}^{n})=1$ (this
can be done since $\mathcal{H}^{2n-s+2\varepsilon}_{\infty}(H)>0$), so that it
satisfies

$\eta(B(x,r))\lesssim(\mathcal{H}^{2n-s+2\varepsilon}_{\infty}(H))^{-1}r^{2n-s+2\varepsilon},$

where the implicit constant depends only on the dimension.

We would like to apply Theorem 2.1 with respect to $\nu_{L}$ and $\eta$.
However, it is not clear if we can do that, since $\nu_{L}$ might not be a
Schwartz function. To overcome this minor issue, we will mollify $\nu_{L}$.

Let $\varphi\mathrel{\mathop{\ordinarycolon}}\mathbb{R}^{n}\to\mathbb{R}$ be a
non-negative radial $C^{\infty}$ function with $\varphi(0)>0,$
$\operatorname{supp}\varphi\subset B(0,1)$ and $\int\varphi=1$. For
$0<\gamma<1$ set $\varphi_{\gamma}(x)=\gamma^{-n}\varphi(x/\gamma)$. Let
$f_{\gamma}=\nu_{L}\ast\varphi_{\gamma}$.

By Theorem 2.1 applied to $f_{\gamma}$ and $\eta$, with $t=2n-s+2\varepsilon$
and $\sigma=(s-n-\varepsilon)/2$, we have

$\int_{\mathbb{R}^{n}}|f_{\gamma}|^{2}\,d\eta\lesssim_{\varepsilon}(\mathcal{H}^{2n-s+2\varepsilon}_{\infty}(H))^{-1}\|f_{\gamma}\|_{H^{\sigma}(\mathbb{R}^{n})}^{2}\\\
\lesssim(\mathcal{H}^{2n-s+2\varepsilon}_{\infty}(H))^{-1}\|\nu_{L}\|_{H^{\sigma}(\mathbb{R}^{n})}^{2},$
(3.4)

where in the last estimate we used the fact that convolving with
$\varphi_{\gamma}$ does not increase the Sobolev norm.

Now we get a lower bound for the left hand side. First, we claim that if
$Q\in\mathcal{B}$ and $0<\gamma\leq\ell(Q)/2$, then

$\int_{Q}f_{\gamma}(x)\,dx\gtrsim M\ell(Q)^{n}.$ (3.5)

Indeed, let $\widetilde{Q}\subset Q$ be a dyadic subcube with
$\ell(\widetilde{Q})=\ell(Q)/2$ and $\nu_{L}(\widetilde{Q})\gtrsim
M\ell(Q)^{n}$ (such subcube exists by (3.1)). It follows from elementary
geometry that the set

$C\coloneqq\\{y\in B(0,\gamma)\ \mathrel{\mathop{\ordinarycolon}}\
y+\widetilde{Q}\subset Q\\}$

is a truncated one-sided cone centered at 0 with opening angle $\sim 1$. Since
$\varphi_{\gamma}$ is radial and $\int_{B(0,\gamma)}\varphi_{\gamma}=1$, we
get $\int_{C}\varphi_{\gamma}(y)\,dy\gtrsim 1$. Thus,

$\int_{Q}f_{\gamma}(x)\,dx=\int_{Q}\nu_{L}\ast\varphi_{\gamma}(x)\,dx=\int_{B(0,\gamma)}\varphi_{\gamma}(y)\int_{Q}\nu_{L}(x-y)\,dx\,dy\\\
\geq\int_{C}\varphi_{\gamma}(y)\int_{Q}\nu_{L}(x-y)\,dx\,dy\geq\int_{C}\varphi_{\gamma}(y)\int_{\widetilde{Q}}\nu_{L}(z)\,dz\,dy\\\
=\nu_{L}(\widetilde{Q})\int_{C}\varphi_{\gamma}(y)\,dy\gtrsim M\ell(Q)^{n},$

where in the last estimate we used that $\nu_{L}(\widetilde{Q})\gtrsim
M\ell(Q)^{n}$, by the definition of $\widetilde{Q}$. This gives (3.5).

It follows from the Cauchy-Schwarz inequality, the property (2.2) of the
Frostman measure $\eta$, and (3.5), that if $Q\in\mathcal{B}$ and
$0<\gamma\leq\ell(Q)/2$, then

$(\eta(Q))^{1/2}\big{(}\int_{Q}|f_{\gamma}|^{2}\,d\eta\big{)}^{1/2}\geq\int_{Q}f_{\gamma}\,d\eta=\frac{\eta(Q)}{\ell(Q)^{n}}\int_{Q}f_{\gamma}(x)\,dx\gtrsim
M\eta(Q),$

and so

$\int_{Q}|f_{\gamma}|^{2}\,d\eta\gtrsim M^{2}\,\eta(Q).$ (3.6)

Let $\mathcal{B}_{\gamma}=\\{Q\in\mathcal{B}\
\mathrel{\mathop{\ordinarycolon}}\ \ell(Q)/2>\gamma\\}$, and
$H_{\gamma}=\bigcup_{Q\in\mathcal{B}_{\gamma}}Q$. Note that
$\\{H_{1/k}\\}_{k\in\mathbb{N}}$ is an increasing sequence of sets, with
$H=\bigcup_{k}H_{1/k}$. Hence, by the continuity of measure we have

$1=\eta(H)=\lim_{k\to\infty}\eta(H_{1/k}).$

Let $k$ be so large that $\eta(H_{1/k})\geq 1/2$. Then, we get from (3.6) and
(3.4) that

$M^{2}\lesssim
M^{2}\sum_{Q\in\mathcal{B}_{1/k}}\eta(Q)\lesssim\int_{H_{1/k}}|f_{1/k}|^{2}\,d\eta\lesssim(\mathcal{H}^{2n-s+2\varepsilon}_{\infty}(H))^{-1}\|\nu_{L}\|_{H^{\sigma}(\mathbb{R}^{n})}^{2}.$

This gives the desired inequality (3.3). ∎

In order to pass from (3.3) to an estimate on the size of
$H^{\prime}=\\{x\in\mathbb{R}^{n}\ \mathrel{\mathop{\ordinarycolon}}\
\mathcal{M}\nu_{L}(x)\geq M\\},$ we will use the well-known one-third trick.
For every $e\in\\{0,1\\}^{n}$ consider the translated dyadic grid on
$\mathbb{R}^{n}$

$\mathbb{D}^{e}_{\mathbb{R}^{n}}=\frac{1}{3}e+\mathbb{D}_{\mathbb{R}^{n}}.$

The proof of the following lemma can be found e.g. in [Ler03, Section 3].

######  Lemma 3.3.

For every $x\in\mathbb{R}^{n}$ and $0<r<1/3$ there exists $e\in\\{0,1\\}^{n}$
and $Q\in\mathbb{D}_{\mathbb{R}^{n}}^{e}$ such that $B(x,r)\subset Q$ and
$\ell(Q)\leq 3r$.

Let $\mathcal{B}^{e}\subset\mathbb{D}^{e}_{\mathbb{R}^{n}}$ be defined
analogously as $\mathcal{B}$, but using the translated grid
$\mathbb{D}^{e}_{\mathbb{R}^{n}}$. Set $H^{e}=\bigcup_{Q\in\mathcal{B}^{e}}Q$.
By Lemma 3.2, for each $e\in\\{0,1\\}^{n}$ we have
$\mathcal{H}^{2n-s+2\varepsilon}_{\infty}(H^{e})\lesssim
M^{-2}\|\nu_{L}\|^{2}_{H^{\sigma}(\mathbb{R}^{n})}$.

######  Lemma 3.4.

We have

$H^{\prime}\subset\bigcup_{e\in\\{0,1\\}^{n}}H^{e}.$

###### Proof.

Let $x\in H^{\prime}$. Then, there exists some ball $B(x,r)$ such that

$\frac{\nu_{L}(B(x,r))}{r^{n}}\geq M.$

Since $M>3^{n}\nu(F)$, we have $0<r<1/3$. By Lemma 3.3, there exists
$e\in\\{0,1\\}^{n}$ and $Q\in\mathbb{D}_{\mathbb{R}^{n}}^{e}$ such that
$B(x,r)\subset Q$ and $\ell(Q)\leq 3r$. Thus,

$\frac{\nu_{L}(Q)}{\ell(Q)^{n}}\geq 3^{-n}\frac{\nu_{L}(B(x,r))}{r^{n}}\geq
3^{-n}M,$

which gives $x\in H^{e}$. ∎

It follows from Lemma 3.4 and Lemma 3.2 that

$\mathcal{H}^{2n-s+2\varepsilon}_{\infty}(H^{\prime})\leq\sum_{e\in\\{0,1\\}^{n}}\mathcal{H}^{2n-s+2\varepsilon}_{\infty}(H^{e})\lesssim_{\varepsilon}M^{-2}\|\nu_{L}\|^{2}_{H^{\sigma}(\mathbb{R}^{n})}.$

Let $\mathcal{Q}\subset\mathbb{D}_{\mathbb{R}^{n}}$ be a family of disjoint
dyadic cubes covering $H^{\prime}$ and such that

$\sum_{Q\in\mathcal{Q}}\ell(Q)^{2n-s+2\varepsilon}\lesssim\mathcal{H}^{2n-s+2\varepsilon}_{\infty}(H^{\prime})\lesssim_{\varepsilon}M^{-2}\|\nu_{\theta}\|^{2}_{H^{\sigma}(\mathbb{R}^{n})}.$
(3.7)

Without loss of generality, we may assume that every $Q\in\mathcal{Q}$
intersects $H^{\prime}$.

For every $Q\in\mathcal{Q}$ let $Q^{\prime}\in\mathbb{D}_{\mathbb{R}^{n}}$ be
the smallest cube with ${Q}^{\prime}\supset Q$ and such that
${Q}^{\prime}\cap(H^{\prime})^{c}\neq\varnothing$, that is, ${Q}^{\prime}$ is
not contained in $H^{\prime}$. Of course, it may happen that $Q^{\prime}=Q$.
Let $\mathcal{P}\subset\mathbb{D}_{\mathbb{R}^{n}}$ be the family of maximal
dyadic cubes from $\\{Q^{\prime}\\}_{Q\in\mathcal{Q}}$. Since $\mathcal{Q}$ is
a covering of $H^{\prime}$, the family $\mathcal{P}$ is also a covering of
$H^{\prime}$. Note that the cubes in $\mathcal{P}$ are pairwise disjoint.

######  Lemma 3.5.

We have

$\sum_{P\in\mathcal{P}}\ell(P)^{2n-s+2\varepsilon}\lesssim_{\varepsilon}M^{-2}\|\nu_{\theta}\|^{2}_{H^{\sigma}(\mathbb{R}^{n})}.$
(3.8)

###### Proof.

We claim that

$\sum_{P\in\mathcal{P}}\ell(P)^{2n-s+2\varepsilon}\lesssim\sum_{Q\in\mathcal{Q}}\ell(Q)^{2n-s+2\varepsilon}.$

The estimate (3.8) will then follow immediately from (3.7).

The bound above is trivial for $\mathcal{P}\cap\mathcal{Q}$, so it suffices to
show

$\sum_{P\in\mathcal{P}\setminus\mathcal{Q}}\ell(P)^{2n-s+2\varepsilon}\lesssim\sum_{Q\in\mathcal{Q}}\ell(Q)^{2n-s+2\varepsilon}.$
(3.9)

Fix a cube $P\in\mathcal{P}\setminus\mathcal{Q}$. It follows that
$P=Q^{\prime}$ for some $Q\in\mathcal{Q}$, and $Q\subsetneq Q^{\prime}$. Since
$Q^{\prime}$ is the smallest ancestor of $Q$ that is not contained in
$H^{\prime}$, there exists a cube
$Q^{\prime\prime}\in\mathbb{D}_{\mathbb{R}^{n}}$ with $Q\subset
Q^{\prime\prime}\subsetneq Q^{\prime}$,
$\ell(Q^{\prime\prime})=\ell(Q^{\prime})/2$, and $Q^{\prime\prime}\subset
H^{\prime}$. At the same time, $\mathcal{Q}$ covers $H^{\prime}$, and in
particular, it covers $Q^{\prime\prime}$. Thus, denoting by $\mathcal{L}^{n}$
the Lebesgue measure on $\mathbb{R}^{n}$,

$\ell(P)^{n}=\ell(Q^{\prime})^{n}=2^{n}\ell(Q^{\prime\prime})^{n}=2^{n}\mathcal{L}^{n}(Q^{\prime\prime})=2^{n}\sum_{R\in\mathcal{Q},\,R\subset
Q^{\prime\prime}}\mathcal{L}^{n}(R)\lesssim\sum_{R\in\mathcal{Q},\,R\subset
P}\ell(R)^{n}.$

Recalling that $0<2n-s+2\varepsilon<n$, we use the elementary inequality
$|\sum{a_{i}}|^{p}\leq\sum|a_{i}|^{p}$ for $p\in(0,1)$ to get

$\ell(P)^{2n-s+2\varepsilon}=\left(\sum_{R\in\mathcal{Q},\,R\subset
P}\ell(R)^{n}\right)^{(2n-s+2\varepsilon)/n}\lesssim\sum_{R\in\mathcal{Q},\,R\subset
P}\ell(R)^{2n-s+2\varepsilon}.$

Summing over $P\in\mathcal{P}\setminus\mathcal{Q}$, and recalling that
$\mathcal{P}$ is a family of disjoint cubes, gives (3.9). ∎

Given a dyadic cube $Q$ and a constant $C\geq 1$, we will write $CQ$ to denote
the cube with the same center as $Q$ and with sidelength $C\ell(Q)$.

######  Lemma 3.6.

For any $P\in{\mathcal{P}}$ and $C\geq 1$ we have

$\nu_{L}(CP)\lesssim_{C}M\ell(P)^{n}.$ (3.10)

###### Proof.

Let $P\in{\mathcal{P}}$. Then, by the definition of $\mathcal{P}$, the cube
$P$ is not contained in $H^{\prime}$. Let $x\in P\setminus H^{\prime}$, and
let $C^{\prime}\sim C$ be such that $B(x,C^{\prime}\ell(P))\supset CP$. Then,

$\frac{\nu_{L}(CP)}{\ell(P)^{n}}\leq(C^{\prime})^{n}\frac{\nu_{L}(B(x,C^{\prime}\ell(P))}{(C^{\prime}\ell(P))^{n}}\leq(C^{\prime})^{n}M,$

where the last inequality follows from the fact that
$\mathcal{M}\nu_{L}(x)\leq M$. ∎

We are ready to conclude the proof of Proposition 3.1. We remark that until
this point we never used lower Ahlfors regularity of measure $\nu$, but now it
will be crucial.

###### Proof of Proposition 3.1.

For every $P\in\mathcal{P}$ let $\mathcal{D}_{P}\subset\mathbb{D}$ be the
family of dyadic cubes $Q\subset[0,1]^{d}$ with $\ell(Q)=\ell(P),\
\pi_{L}(Q)\cap P\neq\varnothing,$ and $F\cap Q\neq\varnothing$. Since
$F_{M}=F\cap\pi_{L}^{-1}(H^{\prime})$, and
$H^{\prime}\subset\bigcup_{P\in\mathcal{P}}P$, we get that

$F_{M}\subset\bigcup_{P\in\mathcal{P}}\bigcup_{Q\in\mathcal{D}_{P}}Q.$

Note that, since $\nu$ is $s$-Ahlfors regular, we have
$\nu(3Q)\sim\ell(Q)^{s}$ for all $Q\in\mathcal{D}_{P}$, the cubes
$\\{3Q\\}_{Q\in\mathcal{D}_{P}}$ have bounded intersections, and they satisfy
$\pi_{L}(3Q)\subset CP$ for some dimensional constant $C\geq 1$. It follows
that

$\mathcal{H}^{n+2\varepsilon}_{\infty}(F_{M})\lesssim\sum_{P\in\mathcal{P}}\sum_{Q\in\mathcal{D}_{P}}\ell(Q)^{n+2\varepsilon}\sim\sum_{P\in\mathcal{P}}\ell(P)^{n+2\varepsilon-s}\sum_{Q\in\mathcal{D}_{P}}\ell(Q)^{s}\\\
\sim\sum_{P\in\mathcal{P}}\ell(P)^{n+2\varepsilon-s}\sum_{Q\in\mathcal{D}_{P}}\nu(3Q)\lesssim\sum_{P\in\mathcal{P}}\ell(P)^{n+2\varepsilon-s}\nu_{L}(CP)\\\
\overset{\eqref{eq:5}}{\lesssim}M\sum_{P\in\mathcal{P}}\ell(P)^{2n+2\varepsilon-s}\overset{\eqref{eq:6}}{\lesssim_{\varepsilon}}M^{-1}\|\nu_{\theta}\|^{2}_{H^{\sigma}(\mathbb{R}^{n})}.$

∎

Theorem 1.3 follows easily from Proposition 3.1.

###### Proof of Theorem 1.3.

Set $\nu=\mathcal{H}^{s}|_{F}$, let $M>3^{n}\nu(F)$ be a large integer. Given
$L\in\mathcal{G}(d,n)$ define

$F_{M}=\\{x\in F\ \mathrel{\mathop{\ordinarycolon}}\
\mathcal{M}\nu_{L}(\pi_{L}(x))\geq M\\}.$

Let $x\in F$ be such that the $(d-n)$-plane $V=V_{x,L}=x+L^{\perp}$ satisfies
$\overline{\dim}_{\mathrm{box}}(V\cap F)>s-n$. We claim that $V\cap F\subset
F_{M}$. Indeed, since $\overline{\dim}_{\mathrm{box}}(V\cap F)>s-n$, there
exists $\beta>0$ and a sequence $\delta_{m}\to 0$ such that

$N(F\cap V,\delta_{m})\geq\delta_{m}^{-s+n-\beta}.$

Since $\nu$ is $s$-Ahlfors regular, it follows that

$\nu(V(2\delta_{m}))\gtrsim\delta_{m}^{-s+n-\beta}\cdot\delta_{m}^{s}=\delta_{m}^{n-\beta},$

where $V(2\delta_{m})$ is the $2\delta_{m}$-neighbourhood of $V$. Hence, for
$x\in V$ we have

$\mathcal{M}\nu_{L}(\pi_{L}(x))\geq\frac{\nu_{L}(B(\pi_{L}(x),2\delta_{m}))}{(2\delta_{m})^{n}}\gtrsim\delta_{m}^{-\beta}.$

If $\delta_{m}$ is chosen small enough, we get $x\in F_{M}$.

The argument above implies that $\\{x\in F\ \mathrel{\mathop{\ordinarycolon}}\
\overline{\dim}_{\mathrm{box}}(F\cap V_{x,L})>s-n\\}\subset F_{M}$. In the
case $s>2n$, it follows from Lemma 2.4 and the Sobolev embedding that
$\nu_{L}\in L^{\infty}(\mathbb{R}^{n})$ for $\gamma_{d,n}$-a.e.
$L\in\mathcal{G}(d,n)$, so that $\mathcal{M}\nu_{L}$ is a bounded function.
Hence, taking $M$ large enough, we get

$\\{x\in F\ \mathrel{\mathop{\ordinarycolon}}\
\overline{\dim}_{\mathrm{box}}(F\cap V_{x,L})>s-n\\}\subset
F_{M}=\varnothing.$ (3.11)

This gives (1.9).

On the other hand, if $n<s\leq 2n$, then by Proposition 3.1

$\mathcal{H}_{\infty}^{n+2\varepsilon}(\\{x\in F\
\mathrel{\mathop{\ordinarycolon}}\ \overline{\dim}_{\mathrm{box}}(F\cap
V_{x,L})>s-n\\})\lesssim_{\varepsilon}M^{-1}\|\nu_{L}\|^{2}_{H^{\sigma}(\mathbb{R}^{n})}.$

Taking $M\to\infty$ gives for all $L\in\mathcal{G}(d,n)$ such that
$\|\nu_{L}\|^{2}_{H^{\sigma}(\mathbb{R}^{n})}<\infty$ (which is true for a.e.
$L\in\mathcal{G}(d,n)$)

$\mathcal{H}_{\infty}^{n+2\varepsilon}(\\{x\in F\
\mathrel{\mathop{\ordinarycolon}}\ \overline{\dim}_{\mathrm{box}}(F\cap
V_{x,L})>s-n\\})=0.$

Letting $\varepsilon\to 0$ finishes the proof. ∎

##  4\. Decomposition of the visible part

We begin the proof of Theorem 1.1. Let $E\subset\mathbb{R}^{d}$ be an
$s$-Ahlfors regular set, with $s\in(d-1,d]$. We set
$\mu=\mathcal{H}^{s}|_{E}$. By rescaling and translating, we may assume that
$\operatorname{diam}(E)\sim 1\sim\mu(E)$ and $E\subset[0,1]^{d}$.

###  4.1. Parameters, cubes, and tubes

Let $\alpha\in(0,1)$ (in the end we will take $\alpha=1-\sqrt{6}/3$), let
$\varepsilon>0$ be a small constant, and set

$\tau\coloneqq\alpha(s-d+1)-5\varepsilon.$ (4.1)

Our goal is to prove that

$\mathcal{H}_{\infty}^{s-\tau}(\operatorname{Vis}_{\theta}(E))=0\quad\text{for
a.e. $\theta\in\mathbb{S}^{d-1}$}.$ (4.2)

Taking $\varepsilon\to 0$, the estimate (1.5) will follow.

Fix a small dyadic scale $\delta\in 2^{-\mathbb{N}}$. In the proof we will
often assume without further mention that $\delta$ is small enough depending
on $\varepsilon$.

We are going to show the following.

######  Proposition 4.1.

For $\alpha=1-\sqrt{6}/3$ we have

$\int_{\mathbb{S}^{d-1}}\mathcal{H}_{\infty}^{s-\tau}(\operatorname{Vis}_{\theta}(E))\,d\theta\lesssim_{\varepsilon}\delta^{\varepsilon}.$

Taking $\delta\to 0$, this will give (4.2).

Let $\Delta\in 2^{-\mathbb{N}}$ be the dyadic number satisfying

$\delta^{\kappa}<\Delta\leq 2\delta^{\kappa},$

where $\kappa\in(0,1)$ is a constant to be fixed later. In the end we will
choose $\kappa=\alpha/(1-\alpha)$. Note that $\Delta\gg\delta$.

Recall that $\mathbb{D}$ stands for the dyadic cubes in $\mathbb{R}^{d}$. We
set

$\displaystyle\mathcal{D}$
$\displaystyle\coloneqq\\{Q\in\mathbb{D}\mathrel{\mathop{\ordinarycolon}}Q\cap
E\neq\varnothing\\},$ $\displaystyle\mathcal{D}_{\delta}$
$\displaystyle\coloneqq\\{Q\in\mathcal{D}\mathrel{\mathop{\ordinarycolon}}\ell(Q)=\delta\\},$
$\displaystyle\mathcal{D}_{\Delta}$
$\displaystyle\coloneqq\\{Q\in\mathcal{D}\mathrel{\mathop{\ordinarycolon}}\ell(Q)=\Delta\\}.$

Since $E$ is $s$-Ahlfors regular, we have
$\\#\mathcal{D}_{\delta}\sim\delta^{-s}$ and
$\\#\mathcal{D}_{\Delta}\sim\delta^{-\kappa s}$.

For every $Q\in\mathcal{D}_{\Delta}$ we define also

$\mu_{Q}=\mu|_{3Q}.$

Note that $\mu_{Q}(\mathbb{R}^{d})=\mu(3Q)\sim\ell(Q)^{s}\sim\delta^{\kappa
s}$.

Recall that $\mu_{\theta}$ denotes the push-forward of $\mu$ by the orthogonal
projection $\pi_{\theta}$, so that it is a measure on
$\theta^{\perp}\simeq\mathbb{R}^{d-1}$.

Let $\sigma=(s-d+1-\varepsilon)/2$. By Lemma 2.4 we have

$\int_{\mathbb{S}^{d-1}}\|\mu_{\theta}\|^{2}_{H^{\sigma}(\mathbb{R}^{d-1})}\,d\theta\lesssim_{\varepsilon}1,$
(4.3)

and for every $Q\in\mathcal{D}_{\Delta}$

$\int_{\mathbb{S}^{d-1}}\|\mu_{Q,\theta}\|^{2}_{H^{\sigma}(\mathbb{R}^{d-1})}\,d\theta\lesssim_{\varepsilon}\mu(3Q)\sim\delta^{\kappa
s}.$ (4.4)

Given $\theta\in\mathbb{S}^{d-1}$ and a dyadic parameter $\gamma\in
2^{-\mathbb{N}}$, let $\mathcal{T}_{\gamma,\theta}$ be a family of
(approximate) tubes given by

$\mathcal{T}_{\gamma,\theta}\coloneqq\\{T\ \mathrel{\mathop{\ordinarycolon}}\
T=\pi_{\theta}^{-1}(Q),\ Q\in\mathbb{D}_{\mathbb{R}^{d-1}},\ \ell(Q)=\gamma,\
T\cap[0,1]^{d}\neq\varnothing\\},$

where $\mathbb{D}_{\mathbb{R}^{d-1}}$ denotes the standard dyadic cubes on
$\mathbb{R}^{d-1}$. When the direction $\theta$ is clear from context, we will
simply write $\mathcal{T}_{\gamma}$.

Note that the tubes from $\mathcal{T}_{\gamma,\theta}$ have width
$\sim\gamma$, they are parallel to $\theta$, they cover $[0,1]^{d}$, and
$\\#\mathcal{T}_{\gamma,\theta}\sim\gamma^{d-1}$.

We set $\mathcal{T}_{\theta}=\bigcup_{\gamma}\mathcal{T}_{\gamma,\theta}$, and
for $T\in\mathcal{T}_{\theta}$ we denote by $w(T)$ the width of $T$, i.e. the
unique $\gamma$ such that $T\in\mathcal{T}_{\gamma}$.

Given a tube $T=\pi_{\theta}^{-1}(Q)\in\mathcal{T}_{\gamma,\theta}$ and a
constant $C>1$, we will write $CT$ to denote $\pi_{\theta}^{-1}(CQ)$.

Let $\theta\in\mathbb{S}^{d-1}$. We will divide
$\operatorname{Vis}_{\theta}(E)$ into four parts. The Hausdorff content of
each will be estimated separately.

###  4.2. Heavy tubes

Since $E$ is an $s$-Ahlfors regular set, we expect that for a typical tube
$T\in\mathcal{T}_{\gamma}=\mathcal{T}_{\gamma,\theta}$

$N(T\cap E,\gamma)\lesssim\gamma^{-(s-d+1)}.$

We are going to bound the number of exceptional tubes where the estimate above
fails badly.

We say that $T\in\mathcal{T}_{\theta}$ is _heavy_ , denoted by
$\mathcal{T}_{H}$, if

$N(T\cap E,\gamma)\geq\delta^{-2\varepsilon}\gamma^{-s+d-1},$ (4.5)

where $\gamma=w(T)$.

Let $Q\in\mathcal{D}_{\Delta}$. We will say that $T\in\mathcal{T}_{\delta}$ is
_heavy inside $Q$_, denoted by $\mathcal{T}_{H,Q}$, if

$N(T\cap E\cap
Q,\delta)\geq\delta^{(\kappa-1)(s-d+1)-\kappa\tau-4\varepsilon}.$ (4.6)

Remark that the definition of $\mathcal{T}_{H}$ includes tubes of varying
widths, whereas the tubes in $\mathcal{T}_{H,Q}$ are all $\delta$-tubes.

We define

$E_{H}^{\prime}\coloneqq E\cap\bigcup_{T\in\mathcal{T}_{H}}T,$

and for each $Q\in\mathcal{D}_{\Delta}$ we define

$Q_{H}\coloneqq 3Q\cap E\cap\bigcup_{T\in\mathcal{T}_{H,Q}}T.$

Finally, we set

$E_{H}\coloneqq E_{H}^{\prime}\cup\bigcup_{Q\in\mathcal{D}_{\Delta}}Q_{H}.$

######  Remark 4.2.

Note that the definitions above depend on the direction $\theta$. To simplify
notation we usually suppress this dependence, but at times we will write
$E_{H,\theta}$ instead of $E_{H}$. The same applies to other sets defined in
this section.

###  4.3. Light tubes

We will say that a tube $T\in\mathcal{T}_{\delta}$ is _light_ if

$N(T\cap E,\delta)\leq\delta^{-(s-d+1)+\tau+\varepsilon}.$

We denote the family of light tubes by $\mathcal{T}_{L}$, and we set

$E_{L}=E\cap\bigcup_{T\in\mathcal{T}_{L}}T.$

######  Lemma 4.3.

We have

$\mathcal{H}^{s-\tau}_{\infty}(E_{L})\lesssim\delta^{\varepsilon}.$

###### Proof.

Observe that

$N(E_{L},\delta)\leq\sum_{T\in\mathcal{T}_{L}}N(T\cap
E,\delta)\lesssim\delta^{1-d}\cdot\delta^{-s+d-1+\tau+\varepsilon}=\delta^{-s+\tau+\varepsilon},$

which gives

$\mathcal{H}_{\infty}^{s-\tau}(E_{L})\leq
N(E_{L},\delta)\cdot\delta^{s-\tau}\lesssim\delta^{\varepsilon}.$

∎

###  4.4. Good and bad parts

Let $Q\in\mathcal{D}_{\Delta}$. We will say that a tube
$T\in\mathcal{T}_{\delta}$ _substantially intersects $Q$_, denoted by
$T\in\mathcal{T}(Q)$, if

$N(T\cap E\cap Q,\,\delta)\geq\delta^{(\kappa-1)(s-d+1)+\tau+4\varepsilon}.$
(4.7)

The following lemma explains where the exponent
$(\kappa-1)(s-d+1)+\tau+4\varepsilon$ came from.

######  Lemma 4.4.

Let $T\in\mathcal{T}_{\delta}$ be a tube which is not contained in any tube
from $\mathcal{T}_{L}\cup\mathcal{T}_{H}$. Then, there exists
$Q\in\mathcal{D}_{\Delta}$ such that $T\in\mathcal{T}(Q)$.

###### Proof.

Let $T^{\prime}\in\mathcal{T}_{\Delta}$ be such that $T\subset T^{\prime}$. By
our assumption, $T^{\prime}\notin\mathcal{T}_{H}$, which gives

$N(T\cap E,\Delta)\leq N(T^{\prime}\cap
E,\Delta)\leq\delta^{-2\varepsilon}\Delta^{-s+d-1}\sim\delta^{-\kappa(s-d+1)-2\varepsilon}.$

At the same time, since $T\notin\mathcal{T}_{L}$, we have $N(T\cap
E,\delta)>\delta^{-(s-d+1)+\tau+\varepsilon}.$ The two inequalities imply that
for some $Q\in\mathcal{D}_{\Delta}$ we have

$N(T\cap E\cap Q,\delta)\gtrsim\frac{N(T\cap E,\delta)}{N(T\cap
E,\Delta)}\gtrsim\delta^{(\kappa-1)(s-d+1)+\tau+3\varepsilon}.$

∎

Let $\mathcal{L}$ denote the set of affine lines parallel to $\theta$. We say
that a line $\ell\in\mathcal{L}$ is _bad with respect to
$Q\in\mathcal{D}_{\Delta}$_ if the $\delta$-tube $T\in\mathcal{T}_{\delta}$
with $\ell\subset T$ satisfies $T\in\mathcal{T}(Q)$, and at the same time

$\ell\cap E\cap\overline{3Q}=\varnothing.$ (4.8)

We denote the collection of bad lines with respect to $Q$ by
$\mathcal{L}_{Q,B}$, and set

$L_{Q,B}=\bigcup_{\ell\in\mathcal{L}_{Q,B}}\ell,\quad
L_{B}=\bigcup_{Q\in\mathcal{D}_{\Delta}}L_{Q,B}.$

We define the bad part of $E$ as

$E_{B}\coloneqq E\cap L_{B}\setminus E_{H},$

and the good part of $E$ as

$E_{G}\coloneqq E\setminus(E_{B}\cup E_{H}\cup E_{L}).$

It is clear that

$\operatorname{Vis}_{\theta}(E)\subset E_{H}\cup E_{L}\cup
E_{B}\cup(E_{G}\cap\operatorname{Vis}_{\theta}(E)).$

In order to estimate the Hausdorff content of
$\operatorname{Vis}_{\theta}(E)$, we will estimate the contents of the four
sets on the right hand side separately. We have already estimated
$\mathcal{H}^{s-\tau}_{\infty}(E_{L})$ in Lemma 4.3. The estimates for
$\mathcal{H}^{s-\tau}_{\infty}(E_{H}),$
$\mathcal{H}^{s-\tau}_{\infty}(E_{B})$, and
$\mathcal{H}^{s-\tau}_{\infty}(E_{G}\cap\operatorname{Vis}_{\theta}(E))$ are
obtained in the next three sections.

##  5\. Heavy part

Recall that $\mathcal{M}$ denotes the Hardy-Littlewood maximal function. We
define

$\widetilde{E}_{H}^{\prime}\coloneqq\\{x\in E\
\mathrel{\mathop{\ordinarycolon}}\
\mathcal{M}\mu_{\theta}(\pi_{\theta}(x))\geq\delta^{-\varepsilon}\\},$

and for each $Q\in\mathcal{D}_{\Delta}$ we define

$\widetilde{Q}_{H}\coloneqq\\{x\in 3Q\cap E\
\mathrel{\mathop{\ordinarycolon}}\
\mathcal{M}\mu_{Q,\theta}(\pi_{\theta}(x))\geq\delta^{\kappa(s-d+1)-\kappa\tau-3\varepsilon}\\}.$

######  Lemma 5.1.

We have $E_{H}^{\prime}\subset\widetilde{E}_{H}^{\prime}$ and
$Q_{H}\subset\widetilde{Q}_{H}$.

###### Proof.

Let $x\in E_{H}^{\prime}$, so that $x\in T\cap E$ for some $T\in\mathcal{T}$
satisfying (4.5). It follows that for some absolute $C>1$

$\mathcal{M}\mu_{\theta}(\pi_{\theta}(x))\geq\frac{\mu_{\theta}(B(\pi_{\theta}(x),Cw(T)))}{(Cw(T))^{d-1}}\gtrsim
w(T)^{-d+1}\mu(2T)\\\ \gtrsim w(T)^{-d+1}N(T\cap E,w(T))\cdot
w(T)^{s}\overset{\eqref{eq:heavytube}}{\geq}\delta^{-2\varepsilon},$

where in the third inequality we used $s$-Ahlfors regularity of $E$. Hence,
$x\in\widetilde{E}_{H}^{\prime}$.

Similarly, if $T\in\mathcal{T}_{\delta}$ satisfies (4.6), then for $x\in T\cap
Q\cap E$

$\mathcal{M}\mu_{Q,\theta}(\pi_{\theta}(x))\gtrsim\delta^{-d+1}\mu_{Q}(2T)\gtrsim\delta^{-d+1}N(T\cap
Q\cap
E,\delta)\cdot\delta^{s}\geq\delta^{\kappa(s-d+1)-\kappa\tau-4\varepsilon},$

which gives $x\in\widetilde{Q}_{H}$. ∎

We use Proposition 3.1 to estimate the size of $E_{H}$.

######  Lemma 5.2.

We have

$\int_{\mathbb{S}^{d-1}}\mathcal{H}_{\infty}^{s-\tau}(E_{H,\theta})\
d\theta\lesssim_{\varepsilon}\delta^{\varepsilon}.$ (5.1)

###### Proof.

Recall that

$E_{H,\theta}=E_{H,\theta}^{\prime}\cup\bigcup_{Q\in\mathcal{D}_{\Delta}}Q_{H,\theta}.$

By Lemma 5.1 and Proposition 3.1 applied with $n=d-1$, $\nu=\mu,\ F=E$ and
$M=\delta^{-\varepsilon}$, we have

$\mathcal{H}_{\infty}^{d-1+2\varepsilon}(E_{H,\theta}^{\prime})\leq\mathcal{H}_{\infty}^{d-1+2\varepsilon}(\widetilde{E}_{H,\theta}^{\prime})\lesssim_{\varepsilon}\delta^{\varepsilon}\|\mu_{\theta}\|_{H^{\sigma}(\mathbb{R}^{d-1})}^{2},$

so that

$\int_{\mathbb{S}^{d-1}}\mathcal{H}_{\infty}^{d-1+2\varepsilon}(E_{H,\theta}^{\prime})\
d\theta\lesssim_{\varepsilon}\delta^{\varepsilon}\int_{\mathbb{S}^{d-1}}\|\mu_{\theta}\|_{H^{\sigma}(\mathbb{R}^{d-1})}^{2}\,d\theta\overset{\eqref{eq:sobolevestmu}}{\lesssim_{\varepsilon}}\delta^{\varepsilon}.$
(5.2)

Since $d-1+2\varepsilon<s-\tau$, this is even better than the estimate for
$\mathcal{H}_{\infty}^{s-\tau}(E_{H,\theta}^{\prime})$ that we need.

We move on to estimating $\mathcal{H}_{\infty}^{s-\tau}(Q_{H,\theta})$. We
apply again Proposition 3.1 with $n=d-1$, $\nu=\mu_{Q},$ $F=3\overline{Q}\cap
E$ and $M=\delta^{\kappa(s-d+1)-\kappa\tau-3\varepsilon}$ to get

$\mathcal{H}_{\infty}^{d-1+2\varepsilon}(Q_{H,\theta})\leq\mathcal{H}_{\infty}^{d-1+2\varepsilon}(\widetilde{Q}_{H,\theta})\lesssim_{\varepsilon}\delta^{\kappa\tau+3\varepsilon-\kappa(s-d+1)}\|\mu_{Q,\theta}\|_{H^{\sigma}(\mathbb{R}^{d-1})}^{2}.$

We use the simple inequality
“$\mathcal{H}^{a}_{\infty}(A)\leq\operatorname{diam}(A)^{a-b}\cdot\mathcal{H}^{b}_{\infty}(A)$
if $a>b$” to estimate

$\mathcal{H}_{\infty}^{s-\tau}(Q_{H,\theta})\lesssim\ell(Q)^{s-\tau-(d-1+2\varepsilon)}\,\mathcal{H}_{\infty}^{d-1+2\varepsilon}(Q_{H,\theta})\\\
\lesssim_{\varepsilon}\delta^{\kappa(s-d+1-\tau-2\varepsilon)-\kappa(s-d+1-\tau)+3\varepsilon}\|\mu_{Q,\theta}\|_{H^{\sigma}(\mathbb{R}^{d-1})}^{2}\\\
=\delta^{(3-2\kappa)\varepsilon}\|\mu_{Q,\theta}\|_{H^{\sigma}(\mathbb{R}^{d-1})}^{2}\leq\delta^{\varepsilon}\|\mu_{Q,\theta}\|_{H^{\sigma}(\mathbb{R}^{d-1})}^{2}.$

Summing over $Q\in\mathcal{D}_{\Delta}$ and integrating over
$\theta\in\mathbb{S}^{d-1}$ yields

$\int_{\mathbb{S}^{d-1}}\sum_{Q\in\mathcal{D}_{\Delta}}\mathcal{H}_{\infty}^{s-\tau}(Q_{H,\theta})\
d\theta\lesssim_{\varepsilon}\delta^{\varepsilon}\sum_{Q\in\mathcal{D}_{\Delta}}\int_{\mathbb{S}^{d-1}}\|\mu_{Q,\theta}\|_{H^{\sigma}(\mathbb{R}^{d-1})}^{2}\,d\theta\\\
\overset{\eqref{eq:sobolevestmuQ}}{\lesssim_{\varepsilon}}\delta^{\varepsilon}\sum_{Q\in\mathcal{D}_{\Delta}}\mu(3Q)\sim\delta^{\varepsilon}.$

Together with (5.2), this finishes the proof of (5.1). ∎

##  6\. Bad part

Fix $\theta\in\mathbb{S}^{d-1}$. Recall that the collection of bad lines with
respect to $Q$ parallel to $\theta$ is denoted by $\mathcal{L}_{Q,B}$,
$L_{Q,B}=\bigcup_{\ell\in\mathcal{L}_{Q,B}}\ell$,
$L_{B}=\bigcup_{Q\in\mathcal{D}_{\Delta}}L_{Q,B}$, and that the bad part was
defined as $E_{B}=E\cap L_{B}\setminus E_{H}$. In this section we estimate
$\mathcal{H}_{\infty}^{s-\tau}(E_{B})$.

######  Lemma 6.1.

Let $\sigma=(s-d+1-\varepsilon)/2$. For every $Q\in\mathcal{D}_{\Delta}$ we
have

$\mathcal{H}_{\infty}^{d-1-\tau}(\pi_{\theta}(L_{Q,B}))\lesssim_{\varepsilon}\delta^{(1-3\alpha-2\kappa)(s-d+1)+5\varepsilon}\|\mu_{Q,\theta}\|_{H^{\sigma}(\mathbb{R}^{d-1})}^{2}.$

###### Proof.

Suppose that $\mathcal{H}_{\infty}^{d-1-\tau}(\pi_{\theta}(L_{Q,B}))>0$,
otherwise there is nothing to prove. It is easy to see that
$\pi_{\theta}(L_{Q,B})\subset\mathbb{R}^{d-1}$ is a Borel set, and so we can
use the usual Frostman’s lemma ([Mat95, Theorem 8.8]) to get a Borel
probability measure $\nu$ with
$\operatorname{supp}\nu\subset\pi_{\theta}(L_{Q,B})$ and

$\nu(B(x,r))\lesssim\mathcal{H}_{\infty}^{d-1-\tau}(\pi_{\theta}(L_{Q,B}))^{-1}r^{d-1-\tau}.$
(6.1)

By the definition of bad lines in $\mathcal{L}_{Q,B}$ (4.8), we have

$\pi_{\theta}(L_{Q,B})\cap\pi_{\theta}(\overline{3Q}\cap E)=\varnothing.$

This means that
$\operatorname{supp}\nu\cap\operatorname{supp}\mu_{Q,\theta}=\varnothing$.
Since the supports are compact, we get that for $\eta>0$ small enough

$\int\mu_{Q,\theta}\ast\varphi_{\eta}\ d\nu=0,$ (6.2)

where $\varphi_{\eta}$ is a smooth mollifier, as in the proof of Lemma 3.2. At
the same time, using Plancherel’s identity

$\int\mu_{Q,\theta}\ast\varphi_{\eta}\
d\nu=\int_{\mathbb{R}^{d-1}}\hat{\varphi}(\eta\xi)\widehat{\mu_{Q,\theta}}(\xi)\overline{\hat{\nu}(\xi)}\
d\xi\\\
\geq\bigg{|}\int_{\mathbb{R}^{d-1}}\hat{\varphi}(C\delta\xi)\hat{\varphi}(\eta\xi)\widehat{\mu_{Q,\theta}}(\xi)\overline{\hat{\nu}(\xi)}\
d\xi\bigg{|}\\\
-\bigg{|}\int_{\mathbb{R}^{d-1}}\big{(}1-\hat{\varphi}(C\delta\xi)\big{)}\hat{\varphi}(\eta\xi)\widehat{\mu_{Q,\theta}}(\xi)\overline{\hat{\nu}(\xi)}\
d\xi\bigg{|}=I_{1}-I_{2},$

where $C\geq 1$ is an absolute constant to be fixed below. Note that by (6.2)
we have $I_{1}=I_{2}$.

We estimate $I_{1}$ from below. Note that, by Plancherel,

$I_{1}=\int\mu_{Q,\theta}\ast\varphi_{C\delta}\ast\varphi_{\eta}\ d\nu.$

Observe that, by the definition of bad lines, and (4.7), for any
$x\in\operatorname{supp}\nu\subset\pi_{\theta}(L_{Q,B})$ the line
$\ell=\pi_{\theta}^{-1}(x)\in\mathcal{L}_{Q,B}$ satisfies $N(\ell(C\delta)\cap
E\cap Q,\delta)\gtrsim\delta^{(\kappa-1)(s-d+1)+\tau+4\varepsilon}$. Recalling
that for $y\in E\cap Q$ we have $\mu_{Q}(B(x,\delta))\sim\delta^{s}$, it
follows that

$\mu_{Q}(\ell(C\delta))\gtrsim\delta^{(\kappa-1)(s-d+1)+\tau+4\varepsilon}\cdot\delta^{s},$

which means that, assuming $C\sim 1$ large enough and $\eta>0$ small enough,

$\mu_{Q,\theta}\ast\varphi_{C\delta}\ast\varphi_{\eta}(x)\gtrsim\delta^{\kappa(s-d+1)+\tau+4\varepsilon}.$

Since $\nu$ is a probability measure, we get that

$I_{1}\gtrsim\delta^{\kappa(s-d+1)+\tau+4\varepsilon}.$ (6.3)

Now we estimate $I_{2}$ from above. Since $\hat{\varphi}$ is a bounded
Lipschitz function, and $\hat{\varphi}(0)=1$, we have for any $0\leq a\leq 1$

$|1-\hat{\varphi}(C\delta\xi)|=|\hat{\varphi}(0)-\hat{\varphi}(C\delta\xi)|\lesssim\min(|\delta\xi|,1)\leq\delta^{a}|\xi|^{a}.$

Thus, by the Cauchy-Schwarz inequality

$I_{2}\lesssim\delta^{a}\int_{\mathbb{R}^{d-1}}|\xi|^{a}|\widehat{\mu_{Q,\theta}}(\xi)||\hat{\nu}(\xi)|\
d\xi\\\
\leq\delta^{a}\left(\int_{\mathbb{R}^{d-1}}|\widehat{\mu_{Q,\theta}}(\xi)|^{2}|\xi|^{s-d+1-\varepsilon}\
d\xi\right)^{1/2}\left(\int_{\mathbb{R}^{d-1}}|\widehat{\nu}(\xi)|^{2}|\xi|^{(2a-s+2(d-1)+\varepsilon)-(d-1)}\
d\xi\right)^{1/2}.$

Since $\nu$ satisfies the Frostman condition (6.1), it follows from the Riesz
energy estimate (2.4) and the identity (2.3) that if
$2a-s+2(d-1)+\varepsilon\leq d-1-\tau-\varepsilon$, then

$\int_{\mathbb{R}^{d-1}}|\widehat{\nu}(\xi)|^{2}|\xi|^{(2a-s+2(d-1)+\varepsilon)-(d-1)}\
d\xi\lesssim_{\varepsilon}\mathcal{H}_{\infty}^{d-1-\tau}(\pi_{\theta}(L_{Q,B}))^{-1}.$

Together with the preceding estimate, we arrive at

$I_{2}\lesssim_{\varepsilon}\delta^{a}\|\mu_{Q,\theta}\|_{H^{\sigma}(\mathbb{R}^{d-1})}\,\mathcal{H}_{\infty}^{d-1-\tau}(\pi_{\theta}(L_{Q,B}))^{-1/2},$

as long as

$0\leq a\leq\frac{(s-d+1)-\tau}{2}-\varepsilon.$

Recalling that $I_{1}=I_{2}$, we compare the estimate of $I_{2}$ with (6.3)
and obtain

$\delta^{a}\|\mu_{Q,\theta}\|_{H^{\sigma}(\mathbb{R}^{d-1})}\,\mathcal{H}_{\infty}^{d-1-\tau}(\pi_{\theta}(L_{Q,B}))^{-1/2}\gtrsim_{\varepsilon}\delta^{\kappa(s-d+1)+\tau+4\varepsilon},$

which is equivalent to

$\mathcal{H}_{\infty}^{d-1-\tau}(\pi_{\theta}(L_{Q,B}))\lesssim_{\varepsilon}\delta^{2a-2\kappa(s-d+1)-2\tau-8\varepsilon}\|\mu_{Q,\theta}\|_{H^{\sigma}(\mathbb{R}^{d-1})}^{2}.$

Choosing $a=\frac{(s-d+1)-\tau}{2}-\varepsilon$ we get

$\mathcal{H}_{\infty}^{d-1-\tau}(\pi_{\theta}(L_{Q,B}))\lesssim_{\varepsilon}\delta^{(1-2\kappa)(s-d+1)-3\tau-10\varepsilon}\|\mu_{Q,\theta}\|_{H^{\sigma}(\mathbb{R}^{d-1})}^{2}\\\
=\delta^{(1-2\kappa-3\alpha)(s-d+1)+5\varepsilon}\|\mu_{Q,\theta}\|_{H^{\sigma}(\mathbb{R}^{d-1})}^{2},$

where we used $\tau=\alpha(s-d+1)-5\varepsilon$. ∎

######  Lemma 6.2.

If $2\kappa+3\alpha\leq 1$, then

$\int_{\mathbb{S}^{d-1}}\mathcal{H}^{s-\tau}_{\infty}(E_{B,\theta})\
d\theta\lesssim_{\varepsilon}\delta^{\varepsilon}.$ (6.4)

###### Proof.

Note that if $\kappa$ and $\alpha$ satisfy $2\kappa+3\alpha\leq 1$, then Lemma
6.1 gives

$\mathcal{H}_{\infty}^{d-1-\tau}(\pi_{\theta}(L_{Q,B}))\lesssim_{\varepsilon}\delta^{5\varepsilon}\|\mu_{Q,\theta}\|_{H^{\sigma}(\mathbb{R}^{d-1})}^{2}.$
(6.5)

Let $\mathcal{P}$ be a family of dyadic cubes in $\mathbb{R}^{d-1}$ that
covers
$\pi_{\theta}(L_{B})=\bigcup_{Q\in\mathcal{D}_{\Delta}}\pi_{\theta}(L_{Q,B})$
and such that

$\sum_{P\in\mathcal{P}}\ell(P)^{d-1-\tau}\lesssim\sum_{Q\in\mathcal{D}_{\Delta}}\mathcal{H}_{\infty}^{d-1-\tau}(\pi_{\theta}(L_{Q,B})).$

Clearly, $\mathcal{P}$ is a covering of $\pi_{\theta}(E_{B,\theta}),$ and
without loss of generality we may assume that each $P\in\mathcal{P}$
intersects $\pi_{\theta}(E_{B,\theta}).$

For each $P\in\mathcal{P}$ let $T_{P}\in\mathcal{T}$ be the tube parallel to
$\theta$ such that $\pi_{\theta}(T_{P})=P$. Clearly,

$E_{B,\theta}\subset\bigcup_{P\in\mathcal{P}}T_{P}\cap E.$

Since $P\cap\pi_{\theta}(E_{B,\theta})\neq\varnothing$, we have $T_{P}\cap
E_{B,\theta}\neq\varnothing$. Recalling that $E_{B,\theta}\cap
E_{H,\theta}=\varnothing$, we see that $T_{P}\cap E\not\subset E_{H,\theta}$.
This means that $T_{P}\notin\mathcal{T}_{H}$, and so by the definition of
heavy tubes (4.5),

$N(T_{P}\cap E,\,\ell(P))\leq\delta^{-2\varepsilon}\ell(P)^{-s+d-1}.$

It follows that

$\mathcal{H}^{s-\tau}_{\infty}(E_{B,\theta})\lesssim\sum_{P\in\mathcal{P}}N(T_{P}\cap
E,\,\ell(P))\cdot\ell(P)^{s-\tau}\\\
\lesssim\delta^{-2\varepsilon}\sum_{P\in\mathcal{P}}\ell(P)^{d-1-\tau}\lesssim\delta^{-2\varepsilon}\sum_{Q\in\mathcal{D}_{\Delta}}\mathcal{H}_{\infty}^{d-1-\tau}(\pi_{\theta}(L_{Q,B})).$

Together with (6.5) this gives

$\mathcal{H}^{s-\tau}_{\infty}(E_{B,\theta})\lesssim\delta^{\varepsilon}\sum_{Q\in\mathcal{D}_{\Delta}}\|\mu_{Q,\theta}\|_{H^{\sigma}(\mathbb{R}^{d-1})}^{2}.$

Integrating the above over $\theta\in\mathbb{S}^{d-1}$ and using (4.4) we get
the desired estimate:

$\int_{\mathbb{S}^{d-1}}\mathcal{H}^{s-\tau}_{\infty}(E_{B,\theta})\
d\theta\lesssim\delta^{\varepsilon}\sum_{Q\in\mathcal{D}_{\Delta}}\int_{\mathbb{S}^{d-1}}\|\mu_{Q,\theta}\|_{H^{\sigma}(\mathbb{R}^{d-1})}^{2}\
d\theta\lesssim_{\varepsilon}\delta^{\varepsilon}\sum_{Q\in\mathcal{D}_{\Delta}}\mu(3Q)\sim\delta^{\varepsilon}.$
(6.6)

∎

##  7\. Good part

In this section we estimate
$\mathcal{H}_{\infty}^{s-\tau}(\operatorname{Vis}_{\theta}(E)\cap E_{G})$.
Recall that $\mathcal{T}_{\delta}$ denotes a collection of $\sim\delta^{1-d}$
tubes of width $\delta$, parallel to $\theta$, which cover $[0,1]^{d}$.

######  Lemma 7.1.

For each $T\in\mathcal{T}_{\delta}$ we have

$N(\operatorname{Vis}_{\theta}(E)\cap E_{G}\cap
T,\delta)\lesssim\delta^{(\alpha-1)(s-d+1)-3\varepsilon}+\delta^{(\kappa-1-\kappa\alpha)(s-d+1)-4\varepsilon}.$

###### Proof.

If $T\cap E_{G}=\varnothing$, then there is nothing to prove, so assume that
$T\cap E_{G}\neq\varnothing$. Since $E_{L}\cap E_{G}=\varnothing$ and
$E_{H}\cap E_{G}=\varnothing$, it follows that $T$ is not contained in any
tube from $\mathcal{T}_{L}\cup\mathcal{T}_{H}$. Hence, Lemma 4.4 implies that
there exists at least one $Q\in\mathcal{D}_{\Delta}$ such that
$T\in\mathcal{T}(Q)$, which means that

$N(T\cap E\cap Q,\,\delta)\geq\delta^{(\kappa-1)(s-d+1)+\tau+4\varepsilon}.$
(7.1)

Furthermore, since $T$ is not contained in any tube from $\mathcal{T}_{H}$, we
have that the family
$\mathcal{D}_{\Delta,T}\coloneqq\\{Q\in\mathcal{D}_{\Delta}\
\mathrel{\mathop{\ordinarycolon}}\ Q\cap T\neq\varnothing\\}$ satisfies

$\\#\mathcal{D}_{\Delta,T}\lesssim\delta^{-\kappa(s-d+1)-2\varepsilon}.$ (7.2)

Let $Q_{T}\in\mathcal{D}_{\Delta,T}$ be the “$\theta$-highest” of all cubes
$Q\in\mathcal{D}_{\Delta}$ satisfying (7.1), in the sense that it maximizes
$\inf\\{x\cdot\theta\mathrel{\mathop{\ordinarycolon}}x\in 3Q\\}$ among all
such cubes.

We partition the family $\mathcal{D}_{\Delta,T}$ into 3 subfamilies:

$\displaystyle\mathcal{D}^{<}_{\Delta,T}$
$\displaystyle\coloneqq\big{\\{}Q\in\mathcal{D}_{\Delta,T}\
\mathrel{\mathop{\ordinarycolon}}\ \sup_{x\in 3Q}\,x\cdot\theta<\inf_{x\in
3Q_{T}}x\cdot\theta\big{\\}},$ $\displaystyle\mathcal{D}^{=}_{\Delta,T}$
$\displaystyle\coloneqq\big{\\{}Q\in\mathcal{D}_{\Delta,T}\
\mathrel{\mathop{\ordinarycolon}}\ \sup_{x\in 3Q}\,x\cdot\theta\geq\inf_{x\in
3Q_{T}}x\cdot\theta\geq\inf_{x\in 3Q}x\cdot\theta\big{\\}},$ (7.3)
$\displaystyle\mathcal{D}^{>}_{\Delta,T}$
$\displaystyle\coloneqq\big{\\{}Q\in\mathcal{D}_{\Delta,T}\
\mathrel{\mathop{\ordinarycolon}}\ \inf_{x\in 3Q}\,x\cdot\theta>\inf_{x\in
3Q_{T}}x\cdot\theta\big{\\}}.$

Roughly speaking, the cubes in $\mathcal{D}^{<}_{\Delta,T}$ lie
“$\theta$-below” $Q_{T}$, the cubes in $\mathcal{D}^{>}_{\Delta,T}$ are
“$\theta$-above” $Q_{T}$, and the cubes in $\mathcal{D}^{=}_{\Delta,T}$ are
“$\theta$-on-the-same-height” as $Q_{T}$. Observe that

$\displaystyle\\#\mathcal{D}^{=}_{\Delta,T}\lesssim 1,$ (7.4)
$\displaystyle\\#\mathcal{D}^{>}_{\Delta,T}\leq\\#\mathcal{D}_{\Delta,T}\overset{\eqref{eq:fewbigcubes}}{\lesssim}\delta^{-\kappa(s-d+1)-2\varepsilon}.$
(7.5)

We will estimate $N(\operatorname{Vis}_{\theta}(E)\cap E_{G}\cap T\cap
Q,\delta)$ for cubes $Q$ in different families separately.

_Cubes from $\mathcal{D}^{<}_{\Delta,T}$_. We claim that for
$Q\in\mathcal{D}^{<}_{\Delta,T}$ we have $\operatorname{Vis}_{\theta}(E)\cap
E_{G}\cap T\cap Q=\varnothing$, so that

$N(\operatorname{Vis}_{\theta}(E)\cap E_{G}\cap T\cap
Q,\delta)=0\quad\text{for $Q\in\mathcal{D}^{<}_{\Delta,T}$.}$ (7.6)

Indeed, suppose that there is some $x\in\operatorname{Vis}_{\theta}(E)\cap
E_{G}\cap T\cap Q$. Let $\ell\in\mathcal{L}$ be the affine line such that
$x\in\ell\subset T$. Since $x\in E_{G}$, we have $x\notin L_{B}$, and in
particular $\ell\notin\mathcal{L}_{Q_{T},B}$, i.e. $\ell$ is not bad with
respect to $Q_{T}$. Recalling that $T\in\mathcal{T}(Q_{T})$, it follows from
the definition of bad lines (4.8) that $\ell\cap
E\cap\overline{3Q_{T}}\neq\varnothing.$

Let $y\in\ell\cap E\cap\overline{3Q_{T}}$. Then, by the definition of
$\mathcal{D}^{<}_{\Delta,T}$ we have $x\cdot\theta<y\cdot\theta$. But the
assumption $x\in\operatorname{Vis}_{\theta}(E)$ implies that for all
$z\in\ell\cap E$ we have $x\cdot\theta>z\cdot\theta$, by the definition of
$\operatorname{Vis}_{\theta}(E)$. We have reached a contradiction.

_Cubes from $\mathcal{D}^{=}_{\Delta,T}$_. Recall that $E_{G}\cap
E_{H}=\varnothing$. In particular, for any $Q\in\mathcal{D}^{=}_{\Delta,T}$ we
have $E_{G}\cap Q_{H}=\varnothing$. There are two possibilities: either
$T\in\mathcal{T}_{H,Q}$, or $T\notin\mathcal{T}_{H,Q}$. In the first case, we
get $Q\cap E\cap T\subset Q_{H}\subset E_{H}$, so that $Q\cap E_{G}\cap
T=\varnothing$. In the latter case, the definition of $\mathcal{T}_{H,Q}$
(4.6) gives

$N(Q\cap E_{G}\cap
T,\delta)\lesssim\delta^{(\kappa-1)(s-d+1)-\kappa\tau-4\varepsilon}\leq\delta^{(\kappa-1-\kappa\alpha)(s-d+1)-4\varepsilon}.$
(7.7)

_Cubes from $\mathcal{D}^{>}_{\Delta,T}$_. Recall that $Q_{T}$ was defined as
the “$\theta$-highest” cube in $\mathcal{D}_{\Delta,T}$ satisfying (7.1). This
means that all the cubes in $\mathcal{D}^{>}_{\Delta,T}$ do not satisfy (7.1),
and so for $Q\in\mathcal{D}^{>}_{\Delta,T}$

$N(T\cap E\cap
Q,\delta)\lesssim\delta^{(\alpha+\kappa-1)(s-d+1)-\varepsilon}.$ (7.8)

We use the three estimates above to conclude that

$N(T\cap
E_{G}\cap\operatorname{Vis}_{\theta}(E),\delta)\leq\sum_{Q\in\mathcal{D}_{\Delta,T}}N(Q\cap
T\cap E_{G}\cap\operatorname{Vis}_{\theta}(E),\delta)\\\
\lesssim\sum_{Q\in\mathcal{D}_{\Delta,T}^{=}}\delta^{(\kappa-1-\kappa\alpha)(s-d+1)-4\varepsilon}+\sum_{Q\in\mathcal{D}_{\Delta,T}^{>}}\delta^{(\alpha+\kappa-1)(s-d+1)-\varepsilon}\\\
\overset{\eqref{eq:few=},\eqref{eq:few>}}{\lesssim}\delta^{(\kappa-1-\kappa\alpha)(s-d+1)-4\varepsilon}+\delta^{-\kappa(s-d+1)-2\varepsilon}\cdot\delta^{(\alpha+\kappa-1)(s-d+1)-\varepsilon}\\\
=\delta^{(\kappa-1-\kappa\alpha)(s-d+1)-4\varepsilon}+\delta^{(\alpha-1)(s-d+1)-3\varepsilon}$

∎

######  Lemma 7.2.

If $\kappa=\alpha/(1-\alpha)$ and $\alpha\in(0,1/2)$, then

$\mathcal{H}_{\infty}^{s-\tau}(\operatorname{Vis}_{\theta}(E)\cap
E_{G})\lesssim\delta^{\varepsilon}.$

###### Proof.

Using the estimate from Lemma 7.1 we get that

$\mathcal{H}_{\infty}^{s-\tau}(\operatorname{Vis}_{\theta}(E)\cap
E_{G})\leq\sum_{T\in\mathcal{T}_{\delta}}N(\operatorname{Vis}_{\theta}(E)\cap
E_{G}\cap T,\delta)\cdot\delta^{s-\tau}\\\
\lesssim\delta^{-d+1}\cdot(\delta^{(\kappa-1-\kappa\alpha)(s-d+1)-4\varepsilon}+\delta^{(\alpha-1)(s-d+1)-3\varepsilon})\cdot\delta^{s-\tau}\\\
=\delta^{(\kappa-\kappa\alpha-\alpha)(s-d+1)+\varepsilon}+\delta^{2\varepsilon}$

Taking $\kappa=\alpha/(1-\alpha),$ which satisfies $\kappa\in(0,1)$ for
$\alpha\in(0,1/2)$, we obtain the desired inequality. ∎

###  7.1. Conclusion of the proof of Theorem 1.1

Putting together the estimates from Lemma 4.3, Lemma 5.2, Lemma 6.2, and Lemma
7.2, we get that

$\int_{\mathbb{S}^{d-1}}\mathcal{H}_{\infty}^{s-\tau}(\operatorname{Vis}_{\theta}(E))\,d\theta\lesssim_{\varepsilon}\delta^{\varepsilon}$

as long as $2\kappa+3\alpha\leq 1,$ $\kappa=\alpha/(1-\alpha)$, and
$\alpha\in(0,1/2)$.

Plugging $\kappa=\alpha/(1-\alpha)$ into $2\kappa+3\alpha\leq 1$ we arrive at

$-3\alpha^{2}+6\alpha-1\leq 0,$

and the largest $\alpha\in(0,1/2)$ which satisfies this inequality is
$\alpha=1-\sqrt{6}/3$. This gives Proposition 4.1. Taking $\delta\to 0$, and
then $\varepsilon\to 0$, Theorem 1.1 follows.

##  8\. General compact sets

In this section we prove Theorem 1.2. Since the proof follows quite closely
that of [Orp22], with the major changes already present in our proof of
Theorem 1.1, we will be brief.

Let $E\subset\mathbb{R}^{d}$ be compact. By rescaling, we may assume that
$E\subset[0,1)^{d}$. Let $\varepsilon>0$ be a small constant, and set

$\tau\coloneqq\frac{1}{6}-5\varepsilon.$

Our goal is to prove that

$\mathcal{H}_{\infty}^{d-\tau}(\operatorname{Vis}_{\theta}(E))=0\quad\text{for
a.e. $\theta\in\mathbb{S}^{d-1}$}.$

By taking $\varepsilon\to 0$, (1.6) will follow.

###  8.1. Preliminaries

Fix a small dyadic scale $\delta\in 2^{-\mathbb{N}}$, and let $\Delta\in
2^{-\mathbb{N}}$ with $\Delta\sim\delta^{\kappa}$, where $\kappa=1/6$, so that
$\Delta\gg\delta$. We will show that

$\int_{\mathbb{S}^{d-1}}\mathcal{H}_{\infty}^{d-\tau}(\operatorname{Vis}_{\theta}(E))\,d\theta\lesssim_{\varepsilon}\delta^{\varepsilon}.$

As before, we set
$\mathcal{D}=\\{Q\in\mathbb{D}\mathrel{\mathop{\ordinarycolon}}Q\cap
E\neq\varnothing\\},$
$\mathcal{D}_{\delta}=\\{Q\in\mathcal{D}\mathrel{\mathop{\ordinarycolon}}\ell(Q)=\delta\\},$
and
$\mathcal{D}_{\Delta}=\\{Q\in\mathcal{D}\mathrel{\mathop{\ordinarycolon}}\ell(Q)=\Delta\\}.$

Let $\mu$ be the Frostman measure on $E$ given by Lemma 2.2, with exponent
$t=d-\tau$, so that

$\mu(B(x,r))\lesssim r^{d-\tau}$ (8.1)

and for all $Q\in\mathcal{D}_{\delta}$

$\mu(\overline{Q})\gtrsim\min(\mathcal{H}_{\infty}^{d-\tau}(E\cap
Q),\ell(Q)^{d}).$ (8.2)

For every $Q\in\mathcal{D}_{\Delta}$ we define $\mu_{Q}=\mu|_{Q}$. Let
$\sigma=(1-\tau-\varepsilon)/2$. By Lemma 2.4 we have

$\int_{\mathbb{S}^{d-1}}\|\mu_{\theta}\|^{2}_{H^{\sigma}(\mathbb{R}^{d-1})}\,d\theta\lesssim_{\varepsilon}1,$
(8.3)

and for every $Q\in\mathcal{D}_{\Delta}$

$\int_{\mathbb{S}^{d-1}}\|\mu_{Q,\theta}\|^{2}_{H^{\sigma}(\mathbb{R}^{d-1})}\,d\theta\lesssim_{\varepsilon}\mu(Q).$
(8.4)

###  8.2. Light part

We say that a cube $Q\in\mathcal{D}_{\delta}$ is light, denoted by
$Q\in\mathcal{D}_{\delta,L}$, if

$\mu(Q)\leq\ell(Q)^{d+\varepsilon}=\delta^{d+\varepsilon}.$

Note that, by (8.2), this implies that

$\mathcal{H}^{d-\tau}_{\infty}(E\cap
Q)\leq\mu(\overline{Q})=\mu(Q)\leq\delta^{d+\varepsilon},$ (8.5)

where the equality in the middle holds because by (8.1) $\mu$ does not charge
$\partial Q$, which is $(d-1)$-dimensional. We set

$E_{0}=E\setminus\bigcup_{Q\in\mathcal{D}_{\delta,L}}Q,$

and

$\mathcal{Q}_{\delta}\coloneqq\mathcal{D}_{\delta}\setminus\mathcal{D}_{\delta,L},$

so that $\mathcal{Q}_{\delta}$ is the collection of all dyadic $\delta$-cubes
intersecting $E_{0}$. Given $Q\in\mathcal{D}_{\Delta}$, we set
$\mathcal{Q}_{\delta}(Q)=\\{P\in\mathcal{Q}_{\delta}\
\mathrel{\mathop{\ordinarycolon}}\ P\subset Q\\}$.

Consider families of tubes $\mathcal{T}_{\delta,\theta}$ as in Subsection 4.1.
We will say that a tube $T\in\mathcal{T}_{\delta,\theta}$ is light if

$\\#\\{P\in\mathcal{Q}_{\delta}\ \mathrel{\mathop{\ordinarycolon}}\ P\cap
T\neq\varnothing\\}\leq\delta^{-1+\tau+\varepsilon}.$

We denote the family of light tubes by $\mathcal{T}_{L,\theta}$. When the
direction $\theta$ is clear from context, we will just write
$\mathcal{T}_{L}$.

We set

$E_{L,\theta}=\bigg{(}E_{0}\cap\bigcup_{T\in\mathcal{T}_{L,\theta}}T\bigg{)}\cup\bigcup_{Q\in\mathcal{D}_{\delta,L}}Q\cap
E.$

######  Lemma 8.1.

We have

$\mathcal{H}^{d-\tau}_{\infty}(E_{L,\theta})\lesssim\delta^{\varepsilon}.$

###### Proof.

We estimate the content of light cubes and light tubes separately. Regarding
the light cubes, we have

$\mathcal{H}^{d-\tau}_{\infty}\big{(}\bigcup_{Q\in\mathcal{D}_{\delta,L}}Q\cap
E\big{)}\leq\sum_{Q\in\mathcal{D}_{\delta,L}}\mathcal{H}^{d-\tau}_{\infty}(Q\cap
E)\overset{\eqref{eq:contentsmall}}{\leq}\delta^{-d}\cdot\delta^{d+\varepsilon}=\delta^{\varepsilon}.$

Concerning tubes, observe that

$N\big{(}E_{0}\cap\bigcup_{T\in\mathcal{T}_{L}}T,\delta\big{)}\leq\sum_{T\in\mathcal{T}_{L}}N(T\cap
E_{0},\delta)\lesssim\delta^{1-d}\cdot\delta^{-1+\tau+\varepsilon}=\delta^{-d+\tau+\varepsilon},$

which gives

$\mathcal{H}_{\infty}^{d-\tau}\big{(}E_{0}\cap\bigcup_{T\in\mathcal{T}_{L}}T\big{)}\leq
N\big{(}E\cap\bigcup_{T\in\mathcal{T}_{L}}T,\delta\big{)}\cdot\delta^{d-\tau}\lesssim\delta^{\varepsilon}.$

∎

###  8.3. Bad part

Fix a direction $\theta\in\mathbb{S}^{d-1}$. Let $Q\in\mathcal{D}_{\Delta}$.
If a tube $T\in\mathcal{T}_{\delta}$ satisfies

$\\#\\{P\in\mathcal{Q}_{\delta}(Q)\ \mathrel{\mathop{\ordinarycolon}}\ P\cap
T\neq\varnothing\\}\geq\delta^{\tau+\kappa-1+2\varepsilon},$ (8.6)

we will write $T\in\mathcal{T}(Q)$.

######  Lemma 8.2.

Let $T\in\mathcal{T}_{\delta}\setminus\mathcal{T}_{L}$. Then, there exists
$Q\in\mathcal{D}_{\Delta}$ such that $T\in\mathcal{T}(Q)$.

###### Proof.

Observe that, trivially,

$\\#\\{Q\in\mathcal{D}_{\Delta}\ \mathrel{\mathop{\ordinarycolon}}\ Q\cap
T\neq\varnothing\\}\lesssim\Delta^{-1}\sim\delta^{-\kappa}.$

At the same time, since $T\notin\mathcal{T}_{L}$, we have
$\\#\\{P\in\mathcal{Q}_{\delta}\ \mathrel{\mathop{\ordinarycolon}}\ P\cap
T\neq\varnothing\\}>\delta^{-1+\tau+\varepsilon}.$ By pigeonholing, we get
that for some $Q\in\mathcal{D}_{\Delta}$

$\\#\\{P\in\mathcal{Q}_{\delta}(Q)\ \mathrel{\mathop{\ordinarycolon}}\ P\cap
T\neq\varnothing\\}\gtrsim\frac{\\#\\{P\in\mathcal{Q}_{\delta}\
\mathrel{\mathop{\ordinarycolon}}\ P\cap
T\neq\varnothing\\}}{\\#\\{Q^{\prime}\in\mathcal{D}_{\Delta}\
\mathrel{\mathop{\ordinarycolon}}\ Q^{\prime}\cap
T\neq\varnothing\\}}\gtrsim\delta^{\kappa+\tau-1+\varepsilon}.$

∎

Let $\mathcal{L}$ denote the set of affine lines parallel to $\theta$. We will
write $\ell\in\mathcal{L}_{Q,B}$ if the $\delta$-tube
$T\in\mathcal{T}_{\delta}$ with $\ell\subset T$ satisfies
$T\in\mathcal{T}(Q)$, and at the same time

$\ell\cap E\cap\overline{Q}=\varnothing.$ (8.7)

We set

$L_{Q,B}=\bigcup_{\ell\in\mathcal{L}_{Q,B}}\ell,\quad
L_{B}=\bigcup_{Q\in\mathcal{D}_{\Delta}}L_{Q,B}.$

We define the bad part of $E$ as

$E_{B}\coloneqq E\cap L_{B}.$

######  Lemma 8.3.

For every $Q\in\mathcal{D}_{\Delta}$ we have

$\mathcal{H}_{\infty}^{d-1-\tau}(\pi_{\theta}(L_{Q,B}))\lesssim\delta^{\varepsilon}\|\mu_{Q,\theta}\|_{H^{\sigma}(\mathbb{R}^{d-1})}^{2}.$

###### Proof.

Arguing exactly the same as in the beginning of the proof of Lemma 6.1, we
arrive at the inequality

$I_{1}=\int\mu_{Q,\theta}\ast\varphi_{C\delta}\ast\varphi_{\eta}\ d\nu\leq
I_{2}\\\
\lesssim\delta^{a}\left(\int_{\mathbb{R}^{d-1}}|\widehat{\mu_{Q,\theta}}(\xi)|^{2}|\xi|^{1-\tau-\varepsilon}\
d\xi\right)^{1/2}\left(\int_{\mathbb{R}^{d-1}}|\widehat{\nu}(\xi)|^{2}|\xi|^{(2a-1+\tau+(d-1)+\varepsilon)-(d-1)}\
d\xi\right)^{1/2}.$ (8.8)

Recalling that $\sigma=(1-\tau-\varepsilon)/2$, and noting that for
$a=1/2-\tau-\varepsilon$ we have

$\int_{\mathbb{R}^{d-1}}|\widehat{\nu}(\xi)|^{2}|\xi|^{(2a-1+\tau+(d-1)+\varepsilon)-(d-1)}\
d\xi\lesssim_{\varepsilon}\mathcal{H}_{\infty}^{d-1-\tau}(\pi_{\theta}(L_{Q,B}))^{-1},$

we get that

$I_{2}\lesssim_{\varepsilon}\delta^{1/2-\tau-\varepsilon}\mathcal{H}_{\infty}^{d-1-\tau}(\pi_{\theta}(L_{Q,B}))^{-1/2}\|\mu_{Q,\theta}\|_{H^{\sigma}(\mathbb{R}^{d-1})}.$

On the other hand, it follows from the definition of bad lines
$\mathcal{L}_{Q,B}$ that for each $\ell\in\mathcal{L}_{Q,B}$

$\mu_{Q}(\ell(C\delta))\gtrsim\delta^{\tau+\kappa-1+2\varepsilon}\cdot\delta^{d+\varepsilon},$

where we also used that $\mu_{Q}(P)\geq\delta^{d+\varepsilon}$ for all
$P\in\mathcal{Q}_{\delta}(Q)$. Hence, for each $x\in\pi_{\theta}(L_{Q,B})$

$\mu_{Q,\theta}\ast\varphi_{C\delta}\ast\varphi_{\eta}(x)\gtrsim\delta^{\tau+\kappa+3\varepsilon}.$

Since $\nu$ is a probability measure, we get
$I_{1}\gtrsim\delta^{\tau+\kappa+3\varepsilon}$. Comparing this with the
estimate for $I_{2}$, we arrive at

$\mathcal{H}_{\infty}^{d-1-\tau}(\pi_{\theta}(L_{Q,B}))\lesssim_{\varepsilon}\delta^{1-2\tau-2\varepsilon}\cdot\delta^{-2\tau-2\kappa-6\varepsilon}\|\mu_{Q,\theta}\|_{H^{\sigma}(\mathbb{R}^{d-1})}^{2}.$

Recalling that $\kappa=1/6$ and $\tau=1/6-5\varepsilon$, we get

$\mathcal{H}_{\infty}^{d-1-\tau}(\pi_{\theta}(L_{Q,B}))\lesssim_{\varepsilon}\delta^{12\varepsilon}\|\mu_{Q,\theta}\|_{H^{\sigma}(\mathbb{R}^{d-1})}^{2}.$

∎

######  Lemma 8.4.

We have

$\int_{\mathbb{S}^{d-1}}\mathcal{H}^{d-\tau}_{\infty}(E_{B,\theta})\
d\theta\lesssim_{\varepsilon}\delta^{\varepsilon}.$ (8.9)

###### Proof.

Since
$\pi_{\theta}(E_{B,\theta})\subset\bigcup_{Q\in\mathcal{D}_{\Delta}}\pi_{\theta}(L_{Q,B})$,
we have

$\mathcal{H}^{d-\tau}_{\infty}(E_{B,\theta})\lesssim\mathcal{H}^{d-1-\tau}_{\infty}(\pi_{\theta}(E_{B,\theta}))\leq\sum_{Q\in\mathcal{D}_{\Delta}}\mathcal{H}_{\infty}^{d-1-\tau}(\pi_{\theta}(L_{Q,B})),$

and so by Lemma 8.3

$\int_{\mathbb{S}^{d-1}}\mathcal{H}^{d-\tau}_{\infty}(E_{B,\theta})\
d\theta\lesssim_{\varepsilon}\delta^{\varepsilon}\sum_{Q\in\mathcal{D}_{\Delta}}\int_{\mathbb{S}^{d-1}}\|\mu_{Q,\theta}\|_{H^{\sigma}(\mathbb{R}^{d-1})}^{2}\,d\theta\overset{\eqref{eq:sobolevestmuQ2}}{\lesssim_{\varepsilon}}\delta^{\varepsilon}\sum_{Q\in\mathcal{D}_{\Delta}}\mu(Q)\sim\delta^{\varepsilon}.$

∎

###  8.4. Good part

Fix $\theta\in\mathbb{S}^{d-1}$. We define the good part of $E$ as

$E_{G}\coloneqq E\setminus(E_{B}\cup E_{L}).$

Recall that $\mathcal{T}_{\delta}$ is a family of $\delta$-tubes parallel to
$\theta$.

######  Lemma 8.5.

For any $T\in\mathcal{T}_{\delta}$ we have

$N(\operatorname{Vis}_{\theta}(E)\cap E_{G}\cap
T,\delta)\lesssim\delta^{\tau-1+2\varepsilon}+\delta^{\kappa-1}.$

###### Proof.

Assume that $T\cap E_{G}\neq\varnothing$, otherwise there is nothing to prove.
Since $E_{G}\cap E_{L}=\varnothing$, we get that $T\notin\mathcal{T}_{L}$.
Hence, by Lemma 8.2, there exists $Q\in\mathcal{D}_{\Delta}$ such that
$T\in\mathcal{T}(Q)$, so that

$\\#\\{P\in\mathcal{Q}_{\delta}(Q)\ \mathrel{\mathop{\ordinarycolon}}\ P\cap
T\neq\varnothing\\}\geq\delta^{\tau+\kappa-1+2\varepsilon}.$ (8.10)

Let $Q_{T}\in\mathcal{D}_{\Delta,T}$ be the “$\theta$-highest” of all cubes
$Q\in\mathcal{D}_{\Delta}$ satisfying (8.10), in the sense that it maximizes
$\inf\\{x\cdot\theta\mathrel{\mathop{\ordinarycolon}}x\in Q\\}$ among all such
cubes. Consider families
$\mathcal{D}^{<}_{\Delta,T},\mathcal{D}^{=}_{\Delta,T},\mathcal{D}^{>}_{\Delta,T}$
defined as in (7.3), except with $3Q,3Q_{T}$ replaced by $Q,Q_{T}$.

For cubes $Q\in\mathcal{D}_{\Delta,T}^{<}$, we argue in the same way as in the
proof of Lemma 7.1 that $N(\operatorname{Vis}_{\theta}(E)\cap E_{G}\cap T\cap
Q,\delta)=0$.

For cubes $Q\in\mathcal{D}_{\Delta,T}^{=}$ we use the trivial bound

$N(\operatorname{Vis}_{\theta}(E)\cap E_{G}\cap T\cap Q,\delta)\leq N(T\cap
Q,\delta)\lesssim\delta^{\kappa-1}$

Finally, for cubes $Q\in\mathcal{D}_{\Delta,T}^{>}$ the inequality (8.10)
fails (this follows from the definition of $Q_{T}$), so that

$N(\operatorname{Vis}_{\theta}(E)\cap E_{G}\cap T\cap Q,\delta)\leq
N(E_{0}\cap T\cap Q,\delta)\lesssim\delta^{\tau+\kappa-1+2\varepsilon},$

where we used the fact that $E_{G}\subset E_{0}$, since $E_{G}\cap
E_{L}=\varnothing$.

Putting the three estimates together, and noting that
$\\#\mathcal{D}_{\Delta,T}^{=}\lesssim 1$ and
$\\#\mathcal{D}_{\Delta,T}^{<}\lesssim\Delta^{-1}\sim\delta^{-\kappa}$, we get

$N(\operatorname{Vis}_{\theta}(E)\cap E_{G}\cap
T,\delta)\lesssim\delta^{\kappa-1}+\delta^{\tau-1+2\varepsilon}.$

∎

######  Lemma 8.6.

We have

$\mathcal{H}_{\infty}^{d-\tau}(\operatorname{Vis}_{\theta}(E)\cap
E_{G})\lesssim\delta^{\varepsilon}.$ (8.11)

###### Proof.

Using Lemma 8.5 we get

$\mathcal{H}_{\infty}^{d-\tau}(\operatorname{Vis}_{\theta}(E)\cap
E_{G})\leq\sum_{T\in\mathcal{T}_{\delta}}N(\operatorname{Vis}_{\theta}(E)\cap
E_{G}\cap T,\delta)\cdot\delta^{d-\tau}\\\
\lesssim\delta^{-d+1}\cdot(\delta^{\tau-1+2\varepsilon}+\delta^{\kappa-1})\cdot\delta^{d-\tau}=\delta^{2\varepsilon}+\delta^{\kappa-\tau}.$

Recalling that $\tau=1/6-5\varepsilon=\kappa-5\varepsilon$, we get (8.11). ∎

Putting together estimates from Lemma 8.1, Lemma 8.4, and Lemma 8.6, we get

$\int_{\mathbb{S}^{d-1}}\mathcal{H}_{\infty}^{d-\tau}(\operatorname{Vis}_{\theta}(E))\,d\theta\lesssim_{\varepsilon}\delta^{\varepsilon}.$

Taking $\delta\to 0$, and then $\varepsilon\to 0$, Theorem 1.2 follows.

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# Encoding sinusoidal functions in hybrid automata formalism

Nikolaos Kekatos
_Verimag Laboratory, University of Grenoble Alpes_
<EMAIL_ADDRESS>
###### Abstract

Hybrid systems can express a plethora of physical phenomena and systems as
they can combine continuous and discrete dynamics. There exist several tools
which enable the reachability analysis of hybrid systems modeled as hybrid
automata. However, these tools exhibit certain limitations in the type of
mathematical operations that they natively support. For example, SpaceEx, a
well established tool in the hybrid verification community, supports the use
of linear ODEs in the flow of each discrete location. Mathematical functions
like alegbraic equations or trigonometric functions have to be encoded as the
solutions of a set of ODEs. In this article, we provide a mechanism to define
sinusoidal functions that is supported by SpaceEx. We also note how certain
Simulink blocks can be translated in hybrid automata.

##  1 Introduction

Herein, we illustrate how we can use linear ODEs to express sinusoids and how
to describe them as hybrid automata. Note that our proposed scheme is suitable
for sine waves (functions of time) and applies to formal verification tools,
such as SpaceEx [4].

##  2 Sinusoids via linear ODEs

The differential equation

$\displaystyle\begin{aligned} \dot{x}&=y\\\ \dot{y}&=-\omega^{2}(x-\mu
t)\end{aligned}$ (1)

where $x$, $y$ are the state variables, $\omega$ and $\mu$ are parameters, and
$t$ describes the time evolution, has the following analytical solution

$\displaystyle y(t)$ $\displaystyle=A\sin(\omega t+\varphi)+\mu$ (2a)
$\displaystyle x(t)$ $\displaystyle=-\frac{A}{\omega}\cos(\omega
t+\varphi)+\mu t.$ (2b)

The initial conditions are $y(0)=A\sin(\varphi)+\mu$, and
$x(0)=-\frac{A}{\omega}\cos(\varphi)$. If we ignore the initial phase, i.e.
$\varphi=0$, the initial conditions become $y(0)=\mu$,
$x(0)=-\frac{A}{\omega}$.

##  3 Sinusoids as hybrid automata

Hybrid automata are a common formalism for modeling hybrid systems and their
semantics can be found at [3]. In this part, we show how to construct hybrid
automata that respect the SX format [1]; a formalism that supports network of
hybrid automata and is used by several reachability tools, such as SpaceEx.
Base Component - Sine. For the sinusoid, we only need one _base component_
that we call sin. We have one _location_ , named loc1, with three variables
$x$, $y$, and $t$. Using the equation 1, the _flow_ is given by111Note that we
opt for $omega*omega$ instead of $\omega^{2}$ to avoid any unexpected error
caused by the SpacEx parser.

$\displaystyle x^{\prime}$ $\displaystyle==y$ $\displaystyle y^{\prime}$
$\displaystyle==-\omega*\omega*(x-\mu*t)$ $\displaystyle t^{\prime}$
$\displaystyle==1$

The three variables $x$, $y$, $t$ are declared as controlled, while the
parameters / constants $\omega$ and $\mu$ are uncontrolled. The variable $y$
is a global variable, i.e. it can be communicated/shared to other components,
whereas $y$ and $t$ are local variables. The hybrid automaton requires no
invariant, reset, guard or synchronization label. Detailed information about
modeling hybrid automata in SpaceEx can be found at [2]. The complete base
component is shown in Figure 1.

Figure 1: Sinusoid encoded as a base component of a hybrid automaton (depicted
in SpaceEx MO.E.)

The initial conditions are selected according to Section 2 to match the
desired behavior of the sinusoidal signal.

Base Component - Clock. We also need a base component that measures the global
time. The name of the base component is clock and it has one variable
$t_{gl}$. The variable is controlled and non-local. The base component has
only one location with flow $\dot{t}_{gl}=1$. The base component is portrayed
in Figure 2.

Figure 2: Clock encoded as a base component of a hybrid automaton (depicted in
SpaceEx MO.E.)

Network Component. For testing purposes, we add a network component that is a
network of two hybrid automata. To do so, we add two binds in the Model
Editor. The user can select the values of the parameters $\omega$ and $\mu$
here. The network component is portrayed in Figure 3.

Figure 3: Network component containing a sinusoid and a clock base component
(depicted in SpaceEx MO.E.)

##  4 Application

In this section, we show the applicability of this transformation via a
numerical example as well as its usability within Simulink.

###  4.1 Numerical Example

Consider that we opt for $\omega=1$, $\mu=2$, $A=0.5$, and $\phi=0$. After
loading the model in the web interface, we need to select the initial states
and the output variables. Note that $x(0)=-\frac{A}{\omega}\cos(\phi)=-0.5$,
$y(0)=A\sin(\varphi)+\mu=2$, and $t(0)=0$. For SpaceEx use, the output
variables are defined in the Output tab, see Figure 4 and the initial states
in the Specification tab, see Figure 5.

Figure 4: Selecting the output states in SpaceEx web interface Figure 5:
Selecting the initial conditions in SpaceEx web interface

Then, we finalize the reachability analysis features in the Options tab by
selecting the STC scenario, flowpipe tolerance 0.01, local time horizon 10s
and max. iterations -1 (fixed point). The reachable set is shown in Figure 6.

$0$$2$$4$$6$$8$$10$$1.4$$1.6$$1.8$$2$$2.2$$2.4$$2.6$

Figure 6: Reachability analysis with SpaceEx web interface (plotted with
TikZ).

###  4.2 Simulink sine wave block

The proposed translation scheme is directly applicable to the Simulink sine
wave block222https://www.mathworks.com/help/simulink/slref/sinewave.html.
Figure 7 shows the options that exist in such a Simulink block. Table 1
presents the relation between the options in Simulink and the corresponding
variables in SpaceEx. The translation of some Simulink blocks into hybrid
automata supported by SpaceEx formalism is shown in [6, 5].

Figure 7: Screenshot of a sine wave block in Simulink Table 1: Translating Simulink parameters into SpaceEx parameters Simulink | SpaceEx
---|---
Amplitude | $A$
Bias | $\mu$
Frequency | $\omega$
Phase | $\phi$
Sample time | –

As it can be observed, the user can directly input the values of the Simulink
block into the corresponding SpaceEx parameters. The initial conditions should
be selected accordingly. Note that we shall opt the sampling time to be 0 as
handle continuous-time systems.

For the aforementioned example, we have $A=10$, $\mu=20$, $\omega=0.5$, and
$\phi=0$. Then, $x(0)=-20$ and $y(0)=20$. Figure 8 shows the reachable set
computed with SpaceEx against the Simulink simulation. The initial conditions
are enlarged by 20% for the reachability computations. It can be observed that
the simulation is included in the flowpipe.

Figure 8: Simulation results (Simulink) in red vs Reachable sets (SpaceEx) in
black. Plot done in MATLAB.

## References

  * [1] Scott Cotton, Goran Frehse, and Olivier Lebeltel.  The SpaceEx modeling language.  http://spaceex.imag.fr/sites/default/files/spaceex_modeling_language_0.pdf, 2010\.
  * [2] Goran Frehse.  An Introduction to SpaceEx v0.8.  http://spaceex.imag.fr/documentation/user-documentation/introduction-spaceex-27, 2010\.
  * [3] Goran Frehse.  An Introduction to Hybrid Automata, Numerical Simulation and Reachability Analysis.  In Formal Modeling and Verification of Cyber-Physical Systems, pages 50–81. Springer, 2015.
  * [4] Goran Frehse, Colas Le Guernic, Alexandre Donzé, Scott Cotton, Rajarshi Ray, Olivier Lebeltel, Rodolfo Ripado, Antoine Girard, Thao Dang, and Oded Maler.  SpaceEx: Scalable verification of hybrid systems.  In CAV Conference, 2011.
  * [5] Nikolaos Kekatos.  Formal Verification of Cyber-Physical Systems in the Industrial Model-Based Design Process.  PhD thesis, 2018.
  * [6] Nikolaos Kekatos, Marcelo Forets, and Goran Frehse.  Constructing verification models of nonlinear simulink systems via syntactic hybridization.  In 2017 IEEE 56th Annual Conference on Decision and Control (CDC), pages 1788–1795. IEEE, 2017.