# On the cardinality of sets in ${\bf R}^{d}$ obeying a slightly obtuse angle
bound∗

Tongseok Lim and Robert J. McCann  Tongseok Lim: Krannert School of Management
Purdue University, West Lafayette, Indiana 47907<EMAIL_ADDRESS>/
<EMAIL_ADDRESS>Robert J. McCann: Department of Mathematics
University of Toronto, Toronto ON Canada<EMAIL_ADDRESS>
###### Abstract.

In this paper we explicitly estimate the number of points in a subset
$A\subseteq{\mathbf{R}}^{d}$ as a function of the maximum angle $\angle A$
that any three of these points form, provided $\angle
A<\theta_{d}:=\arccos(-\frac{1}{d})\in(\pi/2,\pi)$. We also show $\angle
A<\theta_{d}$ ensures that $A$ coincides with the vertex set of a convex
polytope. This study is motivated by a question of Paul Erdős and indirectly
by a conjecture of László Fejes Tóth.

∗The authors thank Sungjin Kim and Balász Gerencsér for fruitful remarks, and
Dmitry Zakharov for directing their attention to his work with Kupavskii [13]
and its connection to [7]. TL is grateful for the support of ShanghaiTech
University, and in addition, to the University of Toronto and its Fields
Institute for the Mathematical Sciences, where parts of this work were
performed. RM acknowledges partial support of his research by the Canada
Research Chairs program and Natural Sciences and Engineering Research Council
of Canada Grant 2015-04383 and 2020-04162. ©2020 by the authors.

_Keywords:_ combinatorial geometry, effective bounds, obtuse angle bounds,
acute sets, cardinality, criterion for convex position, Erdős, Füredi, Danzer,
Grünbaum.

_MSC2010 Classification:_ 52C10 (also 05A20, 05B30, 05D99, 52C35)

##  1\. Introduction

Let us begin with a simple definition.

######  Definition 1.1 (Angle bound).

For $A\subseteq{\mathbf{R}}^{d}$ let $\angle A$ denote the smallest value of
$\theta\in[0,\pi]$, such that no triple of points $x,y,z\in A$ determine an
angle $\angle xyz$ greater than $\theta$, i.e.
$\frac{x-y}{|x-y|}\cdot\frac{z-y}{|z-y|}\geq\cos\theta$ for all $x,y,z\in A$.

For a set $A$, we denote by $|A|\in\\{0,1,...,\infty\\}$ its cardinality. In
the 1950s, Paul Erdős raised the following conjecture [6]:

Conjecture: If $A\subseteq{\mathbf{R}}^{d}$ satisfies $\angle A\leq\pi/2$,
then $|A|\leq 2^{d}$.

The conjectured bound $2^{d}$ is obviously sharp, being achieved by the
vertices of the hypercube in ${\mathbf{R}}^{d}$. For $d=3$ he had advertised
the problem a decade earlier [5], which he claimed [6] lead to an unpublished
solution by Kuiper and Boerdijk. For $d\geq 3$ the conjecture was resolved
affirmatively by Danzer and Grünbaum [3], who established it through a chain
of remarkable inequalities reproduced in [1]. They also asked the following
natural question: if the angle bound $\theta$ is acute, i.e.
$\theta\in[0,\frac{\pi}{2}]$, what would be the optimal upper bound for $|A|$
subject to the strict inequality $\angle A<\theta$? Danzer and Grünbaum had
raised a conjecture on this question which was disproved two decades later by
Erdős and Füredi [7]. Since then the question has remained both interesting
and challenging, as e.g. Gerencsér and Harangi [10], [11] and Aigner and
Ziegler [1] discuss; a significant stream of research has focused on how large
a set $A$ can be while satisfying the strictly acute bound $\angle
A<\frac{\pi}{2}$.

Now what if the angle bound $\theta$ is obtuse, i.e. $\theta\in(\pi/2,\pi)$?
In this case much less appears to be known. It is known however that the set
$A\subseteq{\mathbf{R}}^{d}$ subject to $\angle A\leq\theta$ must be finite;
see e.g. Otsetarova [16]. In the plane, the number of obtuse angles determined
by a set is known to grow as a cubic function of its cardinality [2] [9]. Here
we address the following natural question:

######  Question 1.2.

Given $\theta\in(\pi/2,\pi)$, estimate the smallest $N=N(\theta,d)\in{\bf N}$
(if any exists) such that for all $A\subseteq{\mathbf{R}}^{d}$ satisfying
$\angle A\leq\theta$, we have $|A|\leq N$?

Since $N(\pi,d)=+\infty$, it is not obvious that $\theta<\pi$ makes
$N(\theta,d)$ finite; we describe in an appendix below how this fact can be
inferred by refinining the conclusion of Erdős and Füredi [7, Theorem 4.3],
who used sphere packing and covering asymptotics to derive two-sided bounds on
$N(\theta,d)$ for $\theta$ near $\pi$; their statement is predicated on the
unstated requirement that our $N$ (their $n$) be sufficiently large; we
clarify how large in Theorem A.1. After the current manuscript was first
posted to the arXiv, we also learned that Kupavskii and Zakharov [13] had used
a method inspired by Erdös and Füredi’s to show for each
$\frac{\pi}{2}<\theta<\pi$ the growth of $N(\theta,d)$ to be doubly
exponential in $d$, but with $\theta$ dependent rates which are not estimated
explicitly. In this paper, we use an entirely different approach to give the
first explicit upper bound for $N(\theta,d)$ in the region
$\theta<\theta_{d}$, when

(1.1)
$\theta_{d}:=\arccos\Big{(}-\frac{1}{d}\Big{)}\quad\in\quad(\frac{\pi}{2},\pi)$

shrinks to $\pi/2$ as $d\to\infty$. Instead of relying on the sphere packing
and covering arguments (see Appendix A for more details), we observe that
$\theta:=\angle A<\theta_{d}$ implies the points of $A$ form the vertices of a
convex polytope. Our bound for the number of vertices of this polytope in
terms of $\theta$ and $d$ then follows from the Gauss-Bonnet theorem. This
argument is a higher dimensional version of the planar statement that: if all
angles of a convex polygon are at most $2\pi/3$, then the polygon has at most
six vertices.

In the first circulated draft of our companion work [15], we exploited the
present result to attack a conjecture of Fejes Tóth, concerning the placement
of a large number of lines through the origin so as to maximize the expected
acute angle between them. There instead we were interested in maximizing a
power $\alpha>1$ of the (renormalized) angle between each pair of lines.
Having already established the analogous conjecture in the limiting case
$\alpha=\infty$ [14], we used the present results to extend this conclusion to
large finite values of $\alpha$. In subsequent drafts of [15] however, this
argument has been replaced by a different approach (based in part on an
appendix authored by Bilyk, Glazyrin, Matzke, Park and Vlasiuk) which allows
us to extend our conclusion to the larger range $\alpha>\alpha_{\Delta^{d}}$,
where $\alpha_{\Delta^{d}}<2$ in certain cases.

Changing the ambient space from ${\mathbf{R}}^{d}$ to ${\mathbf{R}}^{d+1}$
hereafter, let $\cal H^{d}$ be the area measure on ${\bf
S}^{d}=\\{x\in{\mathbf{R}}^{d+1}\ |\ |x|=1\\}$, and let $\omega_{d}=\cal
H^{d}({\bf S}^{d})$ denote the total area of ${\bf S}^{d}$, e.g.
$\omega_{1}=2\pi$, $\omega_{2}=4\pi$. For $\eta\in(0,\frac{\pi}{2})$, let
$f_{d}(\eta)$ denote the fraction of ${\bf S}^{d}$ covered by generalized
normals to the cone

(1.2) $C_{u,\eta}:=\\{x\in{\mathbf{R}}^{d+1}\ |\ u\cdot
x\geq|u||x|\cos\eta\\}$

of half-angle $\eta$ around $0\neq u\in{\mathbf{R}}^{d+1}$, so that

$\displaystyle f_{d}(\eta)$ $\displaystyle:=\frac{1}{\omega_{d}}\cal
H^{d}(\\{x\in{\bf S}^{d}\ |\ x\cdot z\leq 0\,\mbox{ for every }z\in
C_{u,\eta}\\})$ (1.3)
$\displaystyle=\frac{\omega_{d-1}}{\omega_{d}}\int_{0}^{\frac{\pi}{2}-\eta}\sin^{d-1}(t)dt$
$\displaystyle=\Big{(}\int_{0}^{\pi}\sin^{d-1}(t)dt\Big{)}^{-1}\int_{0}^{\frac{\pi}{2}-\eta}\sin^{d-1}(t)dt.$

Equivalently, $f_{d}(\eta)$ is the zeroth curvature measure that the cone
$C_{u,\eta}$ assigns to its vertex, in the terminology of Federer [8] [17].

Our result is the following.

######  Theorem 1.3 (Cardinality under possibly obtuse angle bound).

Fix $d\geq 1$ and $0<\theta<\theta_{d+1}$ and
$\eta_{d}(\theta):=\arcsin\Big{(}\frac{\sin(\theta/2)}{\sin(\theta_{d}/2)}\Big{)}\in(0,\frac{\pi}{2})$.
If $A\subseteq{\mathbf{R}}^{d+1}$ satisfies $\angle A\leq\theta$, then (1.1)
and (1.3) yield

(1.4) $|A|\leq\frac{1}{f_{d}(\eta_{d}(\theta))}.$

######  Remark 1.4 (Dimensional monotonicity).

For fixed $\theta$, our cardinality bound $1/f_{d}(\eta_{d}(\theta))$
increases with dimension since $f_{d}(\eta)$ is monotone decreasing with
respect to both $\eta$ and $d$: indeed the difference of averages

$\displaystyle\frac{\partial\log f_{d}(\eta)}{\partial d}=$
$\displaystyle\Big{(}\int_{0}^{\frac{\pi}{2}-\eta}\sin^{d-1}(t)dt\Big{)}^{-1}\int_{0}^{\frac{\pi}{2}-\eta}(\log\sin
t)\sin^{d-1}(t)dt$
$\displaystyle-\Big{(}\int_{0}^{\pi}\sin^{d-1}(t)dt\Big{)}^{-1}\int_{0}^{\pi}(\log\sin
t)\sin^{d-1}(t)dt$

is negative since $t\in[0,\pi]\mapsto\log\sin t$ is symmetric about $\pi/2$
and increasing on $[0,\pi/2]$.

######  Example 1.5 (Explicit bounds in low dimensions).

If $d=1$ then $\eta_{1}(\theta)=\frac{\theta}{2}$ and
$f_{1}(\eta)=\frac{1}{2}-\frac{\eta}{\pi}$ so
$f_{1}(\eta_{1}(\theta))^{-1}=\frac{2\pi}{\pi-\theta}$ yields the known sharp
values $\\{2,3,4,\infty\\}$ corresponding to
$\theta\in\\{0,\frac{\pi}{3},\frac{\pi}{2},\pi\\}$. However
$f_{2}(\eta)=\frac{1}{2}(1-\cos(\frac{\pi}{2}-\eta))$ yields a bound
$f_{2}(\eta_{2}(\frac{\pi}{2}))^{-1}\approx 10.9$ worse than the sharp value
$8$ attained by the vertices of the cube in ${\mathbf{R}}^{3}$.

For non-obtuse sets, the bound (1.4) becomes less and less accurate with
increasing dimension, growing slightly faster than exponentially:

######  Claim 1.6 (Asymptotic bounds as dimension increases).

$\ f_{d}(\eta_{d}(\frac{\pi}{2}))^{-1}<2(\frac{\pi}{2})^{2d-1}e^{(d\log
d)/2}[1+O(d^{-1})].$

This growth, while faster than the known sharp value of $2^{d+1}$, is slow
compared to the doubly exponential growth of $N(\theta,d)$ observed for each
fixed $\theta\in(\frac{\pi}{2},\pi)$ in [13], suggesting a rich range of
intermediate asymptotic behaviour in the narrowing region
$\frac{\pi}{2}<\theta<\theta_{d}$ of the $(\theta,d)$ plane. It is also
smaller that the doubly exponential upper bound obtained in Appendix A using
Erdős-Füredi’s sphere-covering arguments (with ${C_{d}}$ independent of $d$),
which confirms that our bounds represent a significant improvement on the
state of the the art at least in their limited range of validity.

Proof of Claim 1.6. From the identity

$\sin(\theta_{d}/2):=\sqrt{\frac{d+1}{2d}}$

which follows from (1.1) (and which is reasserted below in Theorem 2.3) we
derive

$\displaystyle\eta_{d}(\frac{\pi}{2})$
$\displaystyle=\arcsin\Big{(}\sqrt{1-\frac{1}{d+1}}\Big{)}$
$\displaystyle<\frac{\pi}{2}-\sqrt{2(1-\sqrt{1-(d+1)^{-1}})}$
$\displaystyle=\frac{\pi}{2}-d^{-1/2}(1+O(d^{-1}))$

where $\arcsin(t)<\frac{\pi}{2}-\sqrt{2-2t}$ has been used. Estimating (1.3)
using $\frac{2}{\pi}<t^{-1}\sin(t)<1$ on $t\in(0,\frac{\pi}{2})$ yields

$\displaystyle f_{d}(\eta)$
$\displaystyle>\Big{(}2\int_{0}^{\pi/2}t^{d-1}dt\Big{)}^{-1}\int_{0}^{\frac{\pi}{2}-\eta}(\frac{2}{\pi}t)^{d-1}dt$

whence

$\displaystyle f_{d}(\eta_{d}(\frac{\pi}{2}))^{-1}$
$\displaystyle<2(\frac{\pi}{2})^{2d-1}d^{d/2}[1+O(d^{-1})]$

as desired. QED

In the next section we will prove Theorem 1.3 using two propositions which may
have independent interest. Proposition 2.1 shows that the strict inequality
$\angle A<\theta_{d}$ cannot hold unless no point in
$A\subseteq{\mathbf{R}}^{d}$ is a convex combination of $d+1$ others. On the
other hand, if no point in $A$ is a convex combination of other points, then
$A$ consists precisely of the vertices of a convex polytope. Proposition 2.4
combines a spherical diameter-to-radius inequality [4] with the generalized
Gauss-Bonnet theorem to estimate $|A|$ in terms of the angle bound $\angle
A\leq\theta$ in this case. In an appendix we review the sphere-packing and
covering arguments of Erdős and Füredi for comparison, and show how they can
be extended to sets which need not be too large.

##  2\. Proofs

Let $\operatorname{conv}(A)$ denote the convex hull and $\mathop{\rm int}(A)$
the interior of any subset $A\subseteq{\mathbf{R}}^{d}$.

######  Proposition 2.1 (Angle estimates from an interior point of a simplex).

Let $d\geq 2$ and let
$\\{v_{0},...,v_{d}\\}\subseteq{\mathbf{R}}^{d}\setminus\\{0\\}$ be vertices
of a $d$-dimensional simplex containing the origin. Let
$w_{i}=\frac{v_{i}}{|v_{i}|}$. Then

(2.1) $\min_{0\leq i<j\leq d}w_{i}\cdot w_{j}\leq-\frac{1}{d}$

and equality holds if and only if $\operatorname{conv}\\{w_{0},...,w_{d}\\}$
is a regular simplex.

Proof. The proposition clearly holds for $d=2$: since at least one of the
three angles at the origin must exceed $2\pi/3$ unless
$\\{w_{0},w_{1},w_{2}\\}$ is equilateral. We will proceed by an induction on
dimension $d$. Let $V=\\{v_{0},...,v_{d}\\}$, so that $\operatorname{conv}(V)$
is the $d$-dimensional simplex. If the origin lies on the boundary of
$\operatorname{conv}(V)$, then the induction hypothesis yields $\min_{i\neq
j}w_{i}\cdot w_{j}\leq-\frac{1}{d-1}$. So let us assume $0\in{\rm
int}(\operatorname{conv}(V))$. Let $w_{i}=\frac{v_{i}}{|v_{i}|}$. We claim
$W=\\{w_{0},...,w_{d}\\}$ also forms vertices of a $d$-dimensional simplex
containing the origin in its interior. To see this, observe

$\displaystyle 0\in{\rm int}(\operatorname{conv}(V))$
$\displaystyle\iff\text{for every }i,\ v_{i}=-\sum_{j\neq
i}\lambda_{j}v_{j}\text{ for some }\lambda_{j}>0$ $\displaystyle\iff\text{for
every }i,\ w_{i}=-\sum_{j\neq i}\lambda_{j}w_{j}\text{ for some
}\lambda_{j}>0$ $\displaystyle\iff 0\in{\rm int}(\operatorname{conv}(W))$

which proves the claim, and gives $a_{i}>0$ such that
$\sum_{i=0}^{d}a_{i}w_{i}=0$. Without loss of generality assume
$a_{0}=\min_{i}\\{a_{i}\\}$, $a_{1}=\max_{i}\\{a_{i}\\}$. Set
$b_{i}=\frac{a_{i}}{a_{0}}$, so that $w_{0}=-\sum_{i=1}^{d}b_{i}w_{i}$. Now we
claim:

$\min_{1\leq i<j\leq d}w_{i}\cdot w_{j}\geq-\frac{1}{d}\ \text{ implies }\
w_{0}\cdot w_{1}\leq-\frac{1}{d}.$

To see this, observe

$\displaystyle w_{0}\cdot w_{1}$ $\displaystyle=-\sum_{i=1}^{d}b_{i}w_{1}\cdot
w_{i}$ $\displaystyle\leq-b_{1}+\frac{1}{d}\sum_{i=2}^{d}b_{i}$
$\displaystyle\leq-b_{1}+\frac{1}{d}\sum_{i=2}^{d}b_{1}=-\frac{b_{1}}{d}$
$\displaystyle\leq-\frac{1}{d}$

since $b_{1}\geq 1$. We have shown (2.1), and in view of the above
inequalities, we see that equality holds in (2.1) if and only if $w_{i}\cdot
w_{j}=-\frac{1}{d}$ for all $1\leq i<j\leq d$ and $b_{1}=1$, that is
$\sum_{i=0}^{d}w_{i}=0$. By taking dot product with $w_{j}$, $j\neq 0$ on the
last identity, we conclude that equality holds in (2.1) if and only if
$w_{i}\cdot w_{j}=-\frac{1}{d}$ for all $0\leq i<j\leq d$, that is,
$\operatorname{conv}\\{w_{0},...,w_{d}\\}$ forms a regular $d$-dimensional
simplex. QED

######  Corollary 2.2 (Deciding when points lie in convex position).

Let $d\geq 2$ and $A\subseteq{\mathbf{R}}^{d}$. If $\angle
A<\theta_{d}:=\arccos(-\frac{1}{d})$, then $A$ consists of the vertices of a
convex polytope (not necessarily $d$-dimensional).

Proof. If not, there exist $v\in A$ and $\\{v_{0},...,v_{k}\\}\subseteq
A\setminus\\{v\\}$ which forms vertices of a $k$-dimensional simplex such that
$v\in\operatorname{conv}\\{v_{0},...,v_{k}\\}$ by Carathèodory’s theorem.
Proposition 2.1 then yields $i,j$ such that
$\angle(v_{i},v,v_{j})\geq\theta_{k}\geq\theta_{d}$, a contradiction. QED

To derive the desired cardinality bound for convex polytopes, we use the
following spherical version of Jung’s Theorem [12] relating diameter to radius
bounds in flat space, established on the unit sphere ${\bf S}^{d}$ equipped
with its standard round metric by Dekster [4, Theorem 2]:

######  Theorem 2.3 (Dekster, 1995).

Let $R\in(0,\frac{\pi}{2})$. If $H\subseteq{\bf S}^{d}$ satisfies
$\operatorname*{diam}H\leq 2\arcsin\big{(}\sin(\theta_{d}/2)\sin
R\big{)}\in(0,\pi)$, then $H$ can be contained in a closed ball of radius $R$
in ${\bf S}^{d}$. Here $\sin(\theta_{d}/2)=\sqrt{\frac{d+1}{2d}}$ from (1.1).

######  Proposition 2.4 (Cardinality bound for convex polytopes).

Let $d\geq 1$ and $0<\theta<\theta_{d}$. If $V=\\{v_{1},...,v_{n}\\}$ consists
of the vertices of a convex polytope
$\operatorname{conv}(V)\subseteq{\mathbf{R}}^{d+1}$ with non-empty interior
and satisfies $\angle V\leq\theta$, then $n\leq\frac{1}{f_{d}(\eta)}$ with
$\eta=\eta_{d}(\theta)=\arcsin\Big{(}\frac{\sin(\theta/2)}{\sin(\theta_{d}/2)}\Big{)}\in(0,\frac{\pi}{2})$.

Proof. For $\kappa\in(0,\frac{\pi}{2})$ and $u,v\in{\mathbf{R}}^{d+1}$, use
(1.2) to define a family of cones $v+C_{u,\kappa}\subseteq{\mathbf{R}}^{d+1}$
with vertex $v$, direction $u\neq 0$, and half-angle $\kappa$. Now consider
any of the given vertices $v_{i}$; by translating the polytope we may assume
$v_{i}=0$. By dilating $\lambda V$ with sufficiently large $\lambda>0$ we also
assume that only those edges emanating from $v_{i}=0$ meet ${\bf S}^{d}$, and
set $H\subseteq{\bf S}^{d}$ to be the intersection points of those edges with
${\bf S}^{d}$. Notice $\angle V\leq\theta$ implies
$\operatorname*{diam}H\leq\theta$, hence Dekster’s theorem implies, with
$\eta$ in place of $R$, that $H$ (and hence $V$) is contained in the cone
$v_{i}+C_{u_{i},\eta}$ for some unit vector $u_{i}$. Since the conclusion is
invariant under translations and dilations, we deduce there exists unit
vectors $u_{1},...,u_{n}$ such that

(2.2) $\operatorname{conv}V\subseteq\bigcap_{i=1}^{n}v_{i}+C_{u_{i},\eta}.$

Let $f_{i}$ denote the fraction of ${\bf S}^{d}$ occupied by the generalized
normals to $\operatorname{conv}(V)$ at $v_{i}$:

$f_{i}:=\frac{1}{\omega_{d}}\cal H^{d}(\\{x\in{\bf S}^{d}\ |\
(x-v_{i})\cdot(z-v_{i})\leq 0\,\mbox{ for every
}z\in\operatorname{conv}V\\}).$

Then

$\sum f_{i}=1,$

an intuitive fact which can also be seen as a consequence of, e.g., Federer’s
generalization of the Gauss-Bonnet formula, which asserts that for any convex
body the zeroth curvature measure of the entire convex body is unity,
coinciding with its Euler-Poincaré characteristic [8]; in the case of a convex
polytope $\operatorname{conv}(V)$, the zeroth curvature measure vanishes
except at the vertices of the body, and assigns mass $f_{i}$ to $v_{i}$.

Now by the covering property (2.2), we have

(2.3) $1=\sum_{i=1}^{n}f_{i}\geq\sum_{i=1}^{n}f_{d}(\eta)=nf_{d}(\eta),$

whence $n\leq 1/f_{d}(\eta)$. QED

Proof of Theorem 1.3. Let $A\subseteq{\mathbf{R}}^{d+1}$ satisfy $\angle
A\leq\theta<\theta_{d+1}$. We may assume that $A$ is not contained in any
$d$-dimensional hyperplane, since otherwise we may apply an induction on
dimension using the monotonicity of $d\mapsto f_{d}(\eta_{d}(\theta))$
established in Remark 1.4. Then by Corollary 2.2, $A$ is the set of vertices
of a $(d+1)$-dimensional convex polytope, and the theorem follows from
Proposition 2.4 and $\theta_{d+1}<\theta_{d}$. QED

##  Appendix A Finite-size Erdős and Füredi bounds

Erdős and Füredi relate the quantity we have estimated to sphere-packing and
sphere-covering bounds on the unit sphere ${\bf S}^{d-1}$ in [7, Theorem 4.3].
Since their published statement contains at least one misprint (and some tacit
hypotheses, as observed in [13]), let us recount their argument and refine
their conclusions, so that they apply to all sets, and not only to large ones.
Using our angle bound

$\angle P:=\sup_{x,y,z\in P}\angle xyz,$

let

$\alpha_{d}(n):=\inf_{P\subseteq{\mathbf{R}}^{d}\atop|P|=n}\angle P.$

denote the minimal maximal angle made by $n$ points in ${\mathbf{R}}^{d}$,
e.g. $\alpha_{d}(3)=\pi/3=60^{\circ}$, $\alpha_{3}(4)\approx 109.5^{\circ}$,
etc. Note $P\subseteq P^{\prime}$ implies $\angle P\leq\angle P^{\prime}$,
yielding monotonicity of this minimax $\alpha_{d}(n)\leq\alpha_{d}(n+1)$ with
respect to the size of the sets being considered. This means

$N_{d}(\theta):=\sup_{\alpha_{d}(n)\leq\theta}n$

is a non-decreasing inverse to $\alpha_{d}(n)$ (and in fact agrees with
$N(\theta,d)$ from our Question 1.2). Any upper (respectively lower) bound for
$\alpha_{d}(n)$ can therefore be translated into a lower (respectively upper)
bound for $N_{d}(\theta)$, and vice versa. Theorem 4.3 of [7] asserts

(A.1)
$\frac{\pi}{(\log_{2}n)^{1/(d-1)}}<\pi-\alpha_{d}(n)<\frac{4\pi}{(\log_{2}n)^{1/(d-1)}}$

under the tacit assumption that $n$ needs to be sufficiently large. Inspection
of the proof suggests the intended statement should perhaps have been that

(A.2)
$\frac{{c_{d}}}{(1+\log_{2}n)^{1/(d-1)}}<\pi-\alpha_{d}(n)<\frac{{C_{d}}}{(-1+\log_{2}|n-1|)^{1/(d-1)}},$

where ${c_{d}}$ and ${C_{d}}$ are certain constants independent of $d$, to be
estimated presently, and the lower bound is only asserted for large $n$. Note
(A.2) is equivalent to the bound

(A.3)
$2^{({c_{d}}/(\pi-\theta))^{d-1}-1}<N_{d}(\theta)<1+2^{({C_{d}}/(\pi-\theta))^{d-1}+1}$

while the requirement that $N_{d}(\theta)$ be large translates to $\theta$
being close to $\pi$. For $\theta>\pi-{c_{d}}$ the bounds (A.3) grow doubly
exponentially with dimension $d$. Kupavskii and Zakharov established doubly
exponential bounds for $N_{d}(\theta)$ in the full range
$\frac{\pi}{2}<\theta<\pi$, but their rate constants are not explicit and must
evidently depend on $\theta$ near $\frac{\pi}{2}$ [13].

Let us now recall the arguments which establish (A.2). They begin with two
crucial statements concerning the packing and covering of the unit sphere in
${\mathbf{R}}^{d}$ with balls of radius $\rho/2$: namely, that there exist
positive functions ${c_{d}}(\rho)$ and ${C_{d}}(\rho)$ such that:

(Packing): For all $0<\rho<\bar{\rho}<\frac{\pi}{2}$, there exist more than
$({c_{d}}(\bar{\rho})/\rho)^{d-1}$ (undirected) lines through the origin in
${\mathbf{R}}^{d}$ such that any two of them determine an angle greater than
$\rho$.

(Covering): For all $0<\rho<\bar{\rho}<\pi$, there exist fewer than
$({C_{d}}(\bar{\rho})/\rho)^{d-1}$ (undirected) lines through the origin in
${\mathbf{R}}^{d}$ such that any line determines an angle less than $\rho/2$
with at least one of them.

Clearly it costs no generality to suppose ${C_{d}}(\rho)$ and $-{c_{d}}(\rho)$
to be non-decreasing, and to choose them to have finite non-zero limits
${C_{d}}(0^{+})$ and $-{c_{d}}(0^{+})$. We may take them to be independent of
dimension, or we may take them to be minimal, i.e. to be the actual dimension
and radius dependent packing and covering profiles of ${\bf S}^{d-1}$. Erdős
and Füredi assert more, namely that we can take ${c_{d}}(\rho)=1$ and
${C_{d}}(\rho)=4$. However since they give neither reference or proof we have
not attempted to confirm these precise values, but note that they can only
hold under a smallness hypothesis on $\bar{\rho}$ which they presumably had in
mind, since (Covering) contradicts ${C_{d}}(\rho)=4$ when
$\rho\lesssim\frac{\pi}{2}$ if $d=2$. In any case it makes no difference to
the remainder of their argument. Indeed, if we accept these two facts, the
proof of the lower bound is a simple iterative construction while the upper
bound relies on an elementary lemma from graph theory. We now recall this to
refine the conclusions of [7]. See [7] or [13] for a fuller development of
other parts of the proof.

######  Theorem A.1 (Finite size Erdős-Füredi type bounds).

Fix $d\geq 2$, $n\geq 3$ and $0<\bar{\rho}<\pi$. If
${C_{d}}:[0,\infty)\to(0,\infty)$ satisfies (Covering), then the second
inequality in (A.2) holds with ${C_{d}}={C_{d}}(\bar{\rho})$. If
$\bar{\rho}<\frac{\pi}{2}$ and ${c_{d}}:[0,\infty)\to(0,\infty)$ satisfies
(Packing), then the first inequality in (A.2) holds with
${c_{d}}={c_{d}}(\bar{\rho})$ provided
$n>(d-1)\log_{2}\frac{{c_{d}}(\bar{\rho})}{\bar{\rho}}$.

Proof: Packing bound: Given $0<\rho<\bar{\rho}<\frac{\pi}{2}$, fix
$m>({c_{d}}(\bar{\rho})/\rho)^{d-1}$ and lines $L_{1},\ldots,L_{m}$ such that
any two of them make an angle larger than $\rho$. If we regard them instead as
directed lines, the angle between any two of them is less than $\pi-\rho$.
Take two points $A,B$ parallel to $L_{1}$, and then translate them parallel to
$L_{2}$ far enough that the any vector from the original to the translated
pair almost parallels $L_{2}$. The largest angle between these four points, is
at worst the directed angle between $L_{1}$ and $L_{2}$, hence less than
$\pi-\rho$. Now iterate the construction to obtain $2^{m}$ points having
largest angle less than $\pi-\rho$. This shows

$\alpha_{d}(2^{m})\leq\pi-\rho<\pi-\frac{{c_{d}}(\bar{\rho})}{m^{1/(d-1)}},$

and the same bound extends to $\alpha_{d}(n)$ for all $n\leq 2^{m}$. Thus we
have shown the first inequality in (A.2) holds with
${c_{d}}={c_{d}}(\bar{\rho})$ as soon as
$n>(d-1)\log_{2}\frac{{c_{d}}(\bar{\rho})}{\bar{\rho}}$.

Covering bound: Given $0<\rho<\bar{\rho}<\pi$, fix
$m<({C_{d}}(\bar{\rho})/\rho)^{d-1}$ and lines $L_{1},\ldots,L_{m}$ such that
any other line makes an angle less than $\rho/2$ with one of them. For any set
$P$ with $2^{m}+1$ points, color the complete graph on $P$ by assigning to
each edge $xy$ a color $i$ such that $xy$ is within angle $\rho/2$ of $L_{i}$.
Then a lemma of Szekeres [18] asserts the existence of an odd cycle in some
color. (Alternatively, the last paragraph of the proof of Kupavskii and
Zakharov ’s Proposition 1 [13] gives a self-contained proof of an even simpler
statement which also suffices for the present purpose.) This cycle contains
points $x,y,z$ such that all edges of the triangle $\Delta xyz$ are within
angle $\rho/2$ of the same $L_{i}$. The largest angle in this triangle is at
least $\pi-\rho$. Thus $n\geq 2^{m}+1$ implies

$\alpha_{d}(n)\geq\alpha_{d}(2^{m}+1)\geq\pi-\rho>\pi-\frac{{C_{d}}(\bar{\rho})}{m^{1/(d-1)}}.$

Now letting $\rho\to 0$ yields the second inequality in (A.2) holds with
${C_{d}}={C_{d}}(\bar{\rho})$ for all $m$ and hence all $n$. QED

## References

  * [1] Martin Aigner and Günter M. Ziegler. Proofs from The Book. Sixth edition. Including illustrations by Karl H. Hofmann. Springer, Berlin, 2018.
  * [2] J.H. Conway, T.H. Croft, P. Erdős and M.J.T. Guy.  On the distribution of values of angles determined by coplanar points.  J. London Math. Soc. (2), 19 (1979) 137–143.
  * [3] L. Danzer and B. Grünbaum. On two problems concerning convex bodies by P. Erdős and V.L. Klee. (German) Math. Z. 79 (1962), 95–99.
  * [4] B. V. Dekster.  The Jung theorem for spherical and hyperbolic spaces.  Acta Math. Hungar., 67 (4) (1995), 315–331.
  * [5] Paul Erdős.  Problem 4306.  Amer. Math. Monthly 55 (1948) 431.
  * [6] Paul Erdős.  Some unsolved problems.  Michigan Math. J. 4 (1957) 291–300.
  * [7] P. Erdős and Z. Füredi.  The greatest angle among $n$ points in the $d$-dimensional Euclidean space.  In Combinatorial mathematics (Marseille-Luminy, 1981), Ann. Discrete Math. 17 (1983) 275–283, North-Holland, Amsterdam.
  * [8] Herbert Federer.  Curvature measures.  Trans. Amer. Math. Soc., 93 (1959) 418–491.
  * [9] Ruy Fabila-Monroy, Clemens Huemer and Eulàlia Tramuns.  Note on the number of obtuse angles in point sets.  Internat. J. Comput. Geom. Appl., 24 (2014) 177–181.
  * [10] Balász Gerencsér and Viktor Harangi. Acute sets of exponentially optimal size. Discrete Comput. Geom. 62 (2019), no. 4, 775–780.
  * [11] Balázs Gerencsér and Viktor Harangi. Too acute to be true: the story of acute sets. Amer. Math. Monthly 126 (2019), no. 10, 905–914.
  * [12] Heinrich Jung.  Über die kleinste Kugel, die eine räumliche Figur einschliesst.  J. Reine Angew. Math. 123 (1901) 241–257.
  * [13] Andrey Kupavskii and Dmitriy Zakharov.  The right acute angles problem?  European J. Combin. 89 (2020), 103144, 5 pp.
  * [14] Tongseok Lim and Robert J. McCann.  Maximizing powers of the angle between pairs of points in projective space.  Appl. Math. Optim. 84 (2021) 3217–3227.
  * [15] Tongseok Lim and Robert J. McCann.  Maximizing powers of the angle between pairs of points in projective space.  Preprint (2020).
  * [16] Tanya Otsetarova. Boundaries on the number of points in acute sets. https://math.mit.edu/research/highschool/rsi/documents/2017Otsetarova.pdf
  * [17] Rolf Schneider.  Curvature measures of convex bodies.  Ann. Mat. Pura Appl. (4), 116 (1978) 101–134.
  * [18] Gy. Szekeres.  On an extremum problem in the plane.  Amer. J. Math. 63 (1941) 208–210.


XMM-Newton | 55137 | $0.89\pm 0.55$ | $1.76\pm 0.01$ | $11.1\pm 0.55$ | $44.13\pm 0.001$ | $3.34\pm 0.01$ | $9.97\pm 0.85$ | $43.60\pm 0.001$
XMM-Newton | 55141 | $0.81\pm 0.54$ | $1.76\pm 0.01$ | $12.7\pm 0.59$ | $44.19\pm 0.001$ | $3.22\pm 0.01$ | $8.17\pm 0.88$ | $43.72\pm 0.001$
XMM-Newton | 55145 | $0.60\pm 0.57$ | $1.77\pm 0.01$ | $12.1\pm 0.54$ | $44.17\pm 0.001$ | $3.18\pm 0.02$ | $7.92\pm 0.84$ | $43.70\pm 0.001$
XMM-Newton | 55149 | $0.62\pm 0.52$ | $1.77\pm 0.02$ | $13.2\pm 0.61$ | $44.20\pm 0.001$ | $3.08\pm 0.01$ | $7.88\pm 0.82$ | $43.70\pm 0.001$
XMM-Newton | 55153 | $1.07\pm 0.53$ | $1.76\pm 0.01$ | $11.1\pm 0.60$ | $44.13\pm 0.001$ | $3.27\pm 0.01$ | $7.59\pm 0.80$ | $43.69\pm 0.001$
XMM-Newton | 55155 | $1.05\pm 0.54$ | $1.72\pm 0.01$ | $12.1\pm 0.57$ | $44.18\pm 0.001$ | $3.20\pm 0.01$ | $7.70\pm 0.83$ | $43.69\pm 0.001$
Swift/XRT | 53828 | $3.95\pm 0.55$ | $1.58\pm 0.03$ | $6.26\pm 0.87$ | $43.97\pm 0.007$ | $3.90\pm 0.04$ | $2.29\pm 0.94$ | $43.20\pm 0.016$
Swift/XRT | 54185 | $3.10\pm 0.57$ | $1.03\pm 0.08$ | $3.34\pm 0.88$ | $44.00\pm 0.015$ | $3.05\pm 0.04$ | $5.68\pm 1.05$ | $43.56\pm 0.016$
Swift/XRT | 55127 | $1.30\pm 0.58$ | $1.58\pm 0.09$ | $9.23\pm 0.89$ | $44.14\pm 0.011$ | $2.89\pm 0.05$ | $4.62\pm 0.95$ | $43.50\pm 0.022$
Swift/XRT | 57206 | $0.95\pm 0.59$ | $1.56\pm 0.09$ | $9.86\pm 0.94$ | $44.17\pm 0.007$ | $3.87\pm 0.05$ | $9.47\pm 0.98$ | $43.81\pm 0.090$
Swift/XRT | 57454 | $2.42\pm 0.54$ | $1.60\pm 0.07$ | $10.0\pm 1.05$ | $44.16\pm 0.007$ | $3.27\pm 0.07$ | $5.10\pm 0.99$ | $43.51\pm 0.015$
Swift/XRT | 57965 | $2.74\pm 0.53$ | $1.61\pm 0.08$ | $8.65\pm 1.01$ | $44.08\pm 0.002$ | $3.10\pm 0.05$ | $3.77\pm 1.01$ | $43.38\pm 0.003$
Swift/XRT | 58325 | $0.85\pm 0.53$ | $1.67\pm 0.09$ | $10.0\pm 1.20$ | $44.12\pm 0.006$ | $4.20\pm 0.05$ | $4.00\pm 0.95$ | $43.47\pm 0.013$
Swift/XRT | 58689 | $2.91\pm 0.52$ | $1.52\pm 0.08$ | $7.43\pm 1.04$ | $44.18\pm 0.007$ | $2.71\pm 0.03$ | $5.51\pm 0.99$ | $43.58\pm 0.010$
Swift/XRT | 58949 | $1.52\pm 0.57$ | $1.61\pm 0.09$ | $7.79\pm 0.95$ | $44.16\pm 0.007$ | $3.12\pm 0.03$ | $5.25\pm 0.89$ | $43.64\pm 0.012$
Swift/XRT | 59447 | $0.83\pm 0.55$ | $1.84\pm 0.09$ | $15.3\pm 0.98$ | $44.24\pm 0.007$ | $4.02\pm 0.03$ | $5.46\pm 1.05$ | $43.58\pm 0.017$
Mrk 841
Instrument | MJD | $N_{\rm H}$ | $\Gamma^{\rm PC}$ | $Norm^{PC}$ | $\log(L^{\rm PC})$ | $\Gamma^{\rm SE}$ | $Norm^{SE}$ | $\log(L^{\rm SE})$
|  | $(10^{20}~{}cm^{-2})$ |  | $(10^{-3})$ | (erg/s) |  | $(10^{-3})$ | (erg/s)
XMM-Newton | 51922 | $0.24\pm 0.58$ | $1.87\pm 0.03$ | $4.60\pm 0.57$ | $43.88\pm 0.002$ | $3.75\pm 0.03$ | $2.64\pm 0.55$ | $43.42\pm 0.002$
XMM-Newton | 51923 | $0.51\pm 0.52$ | $1.95\pm 0.03$ | $5.67\pm 0.58$ | $43.94\pm 0.002$ | $4.55\pm 0.03$ | $2.59\pm 0.56$ | $43.51\pm 0.002$
XMM-Newton | 51924 | $3.01\pm 0.54$ | $1.76\pm 0.04$ | $3.58\pm 0.57$ | $43.80\pm 0.003$ | $3.64\pm 0.04$ | $3.44\pm 0.58$ | $43.13\pm 0.002$
XMM-Newton | 53386 | $2.28\pm 0.55$ | $1.39\pm 0.02$ | $1.48\pm 0.56$ | $43.60\pm 0.001$ | $4.08\pm 0.02$ | $0.61\pm 0.57$ | $42.82\pm 0.002$
XMM-Newton | 53568 | $1.80\pm 0.52$ | $1.65\pm 0.05$ | $2.53\pm 0.58$ | $43.71\pm 0.002$ | $4.05\pm 0.04$ | $0.42\pm 0.59$ | $42.86\pm 0.007$
XMM-Newton | 57217 | $0.63\pm 0.51$ | $1.90\pm 0.06$ | $5.03\pm 0.67$ | $43.90\pm 0.005$ | $4.81\pm 0.06$ | $1.20\pm 0.63$ | $43.21\pm 0.007$
Swift/XRT | 54101 | $2.97\pm 0.50$ | $1.34\pm 0.09$ | $1.43\pm 0.52$ | $43.62\pm 0.011$ | $3.09\pm 0.03$ | $1.44\pm 0.51$ | $43.14\pm 0.013$
Swift/XRT | 57223 | $1.50\pm 0.58$ | $1.05\pm 0.09$ | $0.50\pm 0.54$ | $43.81\pm 0.088$ | $2.02\pm 0.07$ | $4.75\pm 0.59$ | $43.14\pm 0.015$
Swift/XRT | 57640 | $0.61\pm 0.51$ | $1.63\pm 0.05$ | $3.37\pm 0.49$ | $43.84\pm 0.010$ | $4.07\pm 0.03$ | $1.70\pm 0.43$ | $43.26\pm 0.016$
Swift/XRT | 57937 | $1.33\pm 0.52$ | $1.91\pm 0.08$ | $4.48\pm 0.40$ | $43.85\pm 0.006$ | $4.37\pm 0.03$ | $0.30\pm 0.41$ | $42.92\pm 0.034$
Swift/XRT | 58299 | $0.57\pm 0.55$ | $1.33\pm 0.08$ | $1.67\pm 0.44$ | $43.70\pm 0.010$ | $2.62\pm 0.04$ | $3.38\pm 0.48$ | $42.25\pm 0.007$
Swift/XRT | 58665 | $3.08\pm 0.56$ | $1.64\pm 0.05$ | $3.23\pm 0.47$ | $43.82\pm 0.006$ | $3.44\pm 0.03$ | $2.09\pm 0.51$ | $43.30\pm 0.010$
Swift/XRT | 59210 | $0.66\pm 0.58$ | $1.34\pm 0.05$ | $2.11\pm 0.43$ | $43.78\pm 0.010$ | $2.76\pm 0.03$ | $3.56\pm 0.55$ | $43.36\pm 0.008$
Swift/XRT | 59402 | $2.93\pm 0.53$ | $1.69\pm 0.08$ | $3.76\pm 0.49$ | $43.86\pm 0.006$ | $3.31\pm 0.02$ | $2.27\pm 0.56$ | $43.34\pm 0.010$
PDS 456
Instrument | MJD | $N_{\rm H}$ | $\Gamma^{\rm PC}$ | $Norm^{PC}$ | $\log(L^{\rm PC})$ | $\Gamma^{\rm SE}$ | $Norm^{SE}$ | $\log(L^{\rm SE})$
|  | $(10^{20}~{}cm^{-2})$ |  | $(10^{-3})$ | (erg/s) |  | $(10^{-3})$ | (erg/s)
XMM-Newton | 54355 | $1.24\pm 0.58$ | $2.07\pm 0.03$ | $2.70\pm 0.57$ | $45.08\pm 0.002$ | $4.59\pm 0.03$ | $2.12\pm 0.59$ | $45.08\pm 0.002$
XMM-Newton | 54357 | $1.78\pm 0.57$ | $2.17\pm 0.03$ | $1.87\pm 0.56$ | $44.91\pm 0.001$ | $5.14\pm 0.04$ | $1.77\pm 0.61$ | $45.12\pm 0.002$
XMM-Newton | 56531 | $5.10\pm 0.61$ | $2.24\pm 0.01$ | $3.90\pm 0.23$ | $45.21\pm 0.001$ | $7.34\pm 0.01$ | $7.25\pm 0.32$ | $46.33\pm 0.001$
XMM-Newton | 56550 | $1.63\pm 0.52$ | $1.81\pm 0.02$ | $1.10\pm 0.44$ | $44.77\pm 0.001$ | $3.91\pm 0.02$ | $1.25\pm 0.62$ | $44.72\pm 0.001$
XMM-Newton | 56555 | $1.78\pm 0.54$ | $1.74\pm 0.03$ | $1.10\pm 0.43$ | $44.79\pm 0.001$ | $3.87\pm 0.02$ | $1.31\pm 0.57$ | $44.74\pm 0.002$
XMM-Newton | 56714 | $2.00\pm 0.57$ | $1.91\pm 0.02$ | $1.01\pm 0.45$ | $44.70\pm 0.001$ | $4.36\pm 0.02$ | $0.89\pm 0.51$ | $44.65\pm 0.003$
XMM-Newton | 57835 | $1.39\pm 0.52$ | $1.58\pm 0.03$ | $0.50\pm 0.59$ | $44.48\pm 0.002$ | $4.67\pm 0.03$ | $0.20\pm 0.57$ | $44.04\pm 0.008$
XMM-Newton | 57837 | $1.67\pm 0.53$ | $1.77\pm 0.04$ | $0.77\pm 0.58$ | $44.63\pm 0.002$ | $3.69\pm 0.04$ | $0.39\pm 0.54$ | $44.19\pm 0.006$
XMM-Newton | 58381 | $1.25\pm 0.58$ | $1.97\pm 0.02$ | $3.65\pm 0.51$ | $45.00\pm 0.001$ | $4.85\pm 0.03$ | $1.48\pm 0.45$ | $44.98\pm 0.002$
XMM-Newton | 58750 | $1.86\pm 0.59$ | $2.21\pm 0.03$ | $1.23\pm 0.59$ | $44.82\pm 0.001$ | $6.14\pm 0.04$ | $0.67\pm 0.45$ | $45.10\pm 0.003$
Swift/XRT | 55022 | $4.90\pm 0.81$ | $2.13\pm 0.03$ | $1.93\pm 0.61$ | $44.93\pm 0.004$ | $6.38\pm 0.04$ | $0.36\pm 0.58$ | $45.75\pm 0.015$
Swift/XRT | 55250 | $5.81\pm 0.84$ | $1.66\pm 0.07$ | $0.78\pm 0.65$ | $44.67\pm 0.011$ | $3.21\pm 0.05$ | $1.08\pm 0.57$ | $44.59\pm 0.014$
Swift/XRT | 57935 | $1.69\pm 0.92$ | $1.99\pm 0.08$ | $0.12\pm 0.52$ | $44.69\pm 0.045$ | $1.99\pm 0.09$ | $1.30\pm 0.94$ | $44.79\pm 0.004$
Swift/XRT | 58374 | $2.78\pm 0.94$ | $2.05\pm 0.09$ | $1.41\pm 0.59$ | $44.80\pm 0.005$ | $2.39\pm 0.04$ | $0.99\pm 0.52$ | $44.59\pm 0.008$
Swift/XRT | 58752 | $4.52\pm 0.82$ | $2.23\pm 0.07$ | $1.54\pm 0.57$ | $45.10\pm 0.006$ | $6.52\pm 0.08$ | $1.36\pm 0.58$ | $44.90\pm 0.026$
Swift/XRT | 59445 | $2.63\pm 0.85$ | $2.26\pm 0.04$ | $1.27\pm 0.59$ | $44.71\pm 0.054$ | $2.23\pm 0.04$ | $1.46\pm 0.57$ | $44.78\pm 0.046$
PKS 0558-504
Instrument | MJD | $N_{\rm H}$ | $\Gamma^{\rm PC}$ | $Norm^{PC}$ | $\log(L^{\rm PC})$ | $\Gamma^{\rm SE}$ | $Norm^{SE}$ | $\log(L^{\rm SE})$
|  | $(10^{20}~{}cm^{-2})$ |  | $(10^{-3})$ | (erg/s) |  | $(10^{-3})$ | (erg/s)
XMM-Newton | 51586 | $0.51\pm 0.58$ | $2.06\pm 0.03$ | $5.41\pm 0.58$ | $45.10\pm 0.004$ | $3.88\pm 0.02$ | $3.30\pm 0.59$ | $44.82\pm 0.004$
XMM-Newton | 51688 | $2.92\pm 1.57$ | $1.97\pm 0.09$ | $3.77\pm 1.88$ | $44.96\pm 0.009$ | $3.85\pm 0.07$ | $4.05\pm 1.57$ | $44.88\pm 0.005$
XMM-Newton | 51827 | $2.18\pm 0.75$ | $2.09\pm 0.04$ | $5.37\pm 0.67$ | $45.19\pm 0.003$ | $5.17\pm 0.04$ | $3.57\pm 0.51$ | $45.09\pm 0.002$
XMM-Newton | 52086 | $1.79\pm 0.54$ | $1.79\pm 0.02$ | $3.45\pm 0.54$ | $44.98\pm 0.007$ | $3.51\pm 0.03$ | $3.26\pm 0.52$ | $44.77\pm 0.005$
XMM-Newton | 52201 | $3.28\pm 0.51$ | $2.17\pm 0.03$ | $12.4\pm 0.58$ | $45.23\pm 0.004$ | $3.43\pm 0.03$ | $6.25\pm 0.55$ | $44.07\pm 0.005$
XMM-Newton | 54718 | $2.02\pm 0.56$ | $2.02\pm 0.01$ | $4.44\pm 0.52$ | $45.02\pm 0.006$ | $3.96\pm 0.01$ | $3.24\pm 0.52$ | $44.82\pm 0.005$
XMM-Newton | 54720 | $2.45\pm 0.54$ | $2.07\pm 0.02$ | $6.06\pm 0.53$ | $45.15\pm 0.001$ | $4.43\pm 0.01$ | $3.39\pm 0.53$ | $44.92\pm 0.005$
XMM-Newton | 54722 | $1.81\pm 0.53$ | $2.02\pm 0.02$ | $4.93\pm 0.54$ | $45.07\pm 0.001$ | $4.19\pm 0.01$ | $2.96\pm 0.55$ | $44.82\pm 0.006$
XMM-Newton | 54726 | $1.76\pm 0.53$ | $2.04\pm 0.01$ | $6.23\pm 0.57$ | $45.16\pm 0.001$ | $4.28\pm 0.02$ | $3.77\pm 0.59$ | $44.94\pm 0.005$
Swift/XRT | 54740 | $2.01\pm 0.59$ | $1.84\pm 0.03$ | $4.27\pm 0.51$ | $45.06\pm 0.003$ | $3.72\pm 0.03$ | $3.76\pm 0.58$ | $44.86\pm 0.003$
Swift/XRT | 54921 | $1.91\pm 0.57$ | $2.01\pm 0.02$ | $7.42\pm 0.52$ | $45.25\pm 0.003$ | $4.65\pm 0.02$ | $5.69\pm 0.54$ | $45.19\pm 0.003$
Swift/XRT | 55105 | $1.49\pm 0.52$ | $2.07\pm 0.02$ | $6.59\pm 0.58$ | $45.18\pm 0.003$ | $5.11\pm 0.02$ | $4.53\pm 0.54$ | $45.18\pm 0.004$
Swift/XRT | 55242 | $1.04\pm 0.55$ | $1.98\pm 0.03$ | $7.88\pm 0.57$ | $45.28\pm 0.004$ | $4.42\pm 0.03$ | $6.37\pm 0.59$ | $45.19\pm 0.005$
Swift/XRT | 57711 | $5.91\pm 0.51$ | $2.17\pm 0.05$ | $5.57\pm 0.59$ | $45.07\pm 0.013$ | $4.86\pm 0.05$ | $2.31\pm 0.61$ | $45.83\pm 0.019$
SWIFT J0501.9-3239
Instrument | MJD | $N_{\rm H}$ | $\Gamma^{\rm PC}$ | $Norm^{PC}$ | $\log(L^{\rm PC})$ | $\Gamma^{\rm SE}$ | $Norm^{SE}$ | $\log(L^{\rm SE})$
|  | $(10^{20}~{}cm^{-2})$ |  | $(10^{-3})$ | (erg/s) |  | $(10^{-3})$ | (erg/s)
XMM-Newton | 55225 | $0.92\pm 0.45$ | $1.28\pm 0.02$ | $1.18\pm 0.49$ | $42.63\pm 0.001$ | $4.90\pm 0.02$ | $0.32\pm 0.49$ | $41.87\pm 0.002$
XMM-Newton | 57655 | $0.26\pm 0.42$ | $1.34\pm 0.01$ | $1.05\pm 0.52$ | $42.55\pm 0.001$ | $3.08\pm 0.02$ | $0.53\pm 0.48$ | $41.75\pm 0.002$
Swift/XRT | 53686 | $0.77\pm 0.43$ | $1.25\pm 0.02$ | $1.35\pm 0.55$ | $42.83\pm 0.010$ | $3.65\pm 0.03$ | $0.62\pm 0.41$ | $42.83\pm 0.020$
Swift/XRT | 54476 | $0.16\pm 0.41$ | $1.12\pm 0.08$ | $1.34\pm 0.89$ | $42.84\pm 0.023$ | $2.69\pm 0.07$ | $3.12\pm 0.45$ | $42.56\pm 0.017$
Swift/XRT | 57853 | $0.46\pm 0.46$ | $1.28\pm 0.12$ | $1.17\pm 1.09$ | $42.76\pm 0.027$ | $2.44\pm 0.10$ | $1.32\pm 0.89$ | $42.23\pm 0.034$
NGC 7469
Instrument | MJD | $N_{\rm H}$ | $\Gamma^{\rm PC}$ | $Norm^{PC}$ | $\log(L^{\rm PC})$ | $\Gamma^{\rm SE}$ | $Norm^{SE}$ | $\log(L^{\rm SE})$
|  | $(10^{20}~{}cm^{-2})$ |  | $(10^{-3})$ | (erg/s) |  | $(10^{-3})$ | (erg/s)
XMM-Newton | 51940 | $0.70\pm 0.47$ | $1.80\pm 0.05$ | $7.07\pm 0.58$ | $43.38\pm 0.002$ | $4.12\pm 0.05$ | $4.86\pm 0.52$ | $42.99\pm 0.002$
XMM-Newton | 51941 | $0.80\pm 0.46$ | $1.82\pm 0.04$ | $7.73\pm 0.59$ | $43.41\pm 0.004$ | $4.14\pm 0.05$ | $6.95\pm 0.83$ | $43.05\pm 0.006$
XMM-Newton | 53339 | $0.77\pm 0.49$ | $1.85\pm 0.01$ | $8.52\pm 0.57$ | $43.44\pm 0.003$ | $4.30\pm 0.02$ | $6.23\pm 0.84$ | $43.12\pm 0.005$
XMM-Newton | 53076 | $0.84\pm 0.45$ | $1.81\pm 0.02$ | $8.42\pm 0.56$ | $43.45\pm 0.005$ | $4.32\pm 0.02$ | $5.45\pm 0.54$ | $43.07\pm 0.009$
XMM-Newton | 57185 | $0.60\pm 0.42$ | $1.86\pm 0.02$ | $8.66\pm 0.53$ | $43.44\pm 0.004$ | $4.04\pm 0.02$ | $5.93\pm 0.53$ | $43.07\pm 0.005$
XMM-Newton | 57350 | $0.10\pm 0.43$ | $1.79\pm 0.01$ | $7.86\pm 0.54$ | $43.43\pm 0.003$ | $3.47\pm 0.01$ | $3.28\pm 0.55$ | $42.77\pm 0.006$
XMM-Newton | 57371 | $0.71\pm 0.44$ | $1.76\pm 0.01$ | $6.04\pm 0.55$ | $43.33\pm 0.002$ | $4.04\pm 0.01$ | $4.60\pm 0.51$ | $42.96\pm 0.007$
XMM-Newton | 57379 | $0.62\pm 0.49$ | $1.76\pm 0.01$ | $6.51\pm 0.58$ | $43.36\pm 0.001$ | $3.88\pm 0.02$ | $4.60\pm 0.52$ | $42.95\pm 0.003$
XMM-Newton | 57380 | $0.53\pm 0.41$ | $1.77\pm 0.02$ | $7.24\pm 0.57$ | $43.41\pm 0.005$ | $3.67\pm 0.03$ | $5.48\pm 0.57$ | $43.01\pm 0.008$
XMM-Newton | 57382 | $0.61\pm 0.47$ | $1.76\pm 0.02$ | $7.24\pm 0.59$ | $43.41\pm 0.006$ | $4.00\pm 0.02$ | $4.33\pm 0.58$ | $42.93\pm 0.009$
XMM-Newton | 57384 | $0.50\pm 0.40$ | $1.77\pm 0.02$ | $7.80\pm 0.50$ | $43.44\pm 0.002$ | $3.37\pm 0.02$ | $3.27\pm 0.58$ | $42.77\pm 0.007$
Swift/XRT | 53899 | $0.10\pm 0.61$ | $1.23\pm 0.04$ | $1.56\pm 0.68$ | $43.01\pm 0.016$ | $2.50\pm 0.04$ | $5.69\pm 0.74$ | $42.88\pm 0.001$
Swift/XRT | 54704 | $0.61\pm 0.67$ | $1.54\pm 0.02$ | $2.54\pm 0.67$ | $43.05\pm 0.005$ | $3.22\pm 0.03$ | $2.70\pm 0.73$ | $42.69\pm 0.007$
Swift/XRT | 56528 | $0.42\pm 0.65$ | $1.84\pm 0.01$ | $7.44\pm 0.61$ | $43.39\pm 0.004$ | $3.09\pm 0.02$ | $2.48\pm 0.71$ | $42.66\pm 0.022$
Swift/XRT | 56462 | $0.12\pm 0.60$ | $1.60\pm 0.01$ | $2.94\pm 0.62$ | $43.09\pm 0.005$ | $2.45\pm 0.02$ | $2.14\pm 0.72$ | $42.67\pm 0.009$
Swift/XRT | 56491 | $0.11\pm 0.61$ | $1.73\pm 0.02$ | $2.42\pm 0.63$ | $43.94\pm 0.006$ | $2.48\pm 0.02$ | $0.63\pm 0.74$ | $42.52\pm 0.024$
Swift/XRT | 56514 | $0.08\pm 0.60$ | $1.76\pm 0.02$ | $4.59\pm 0.68$ | $43.21\pm 0.005$ | $3.72\pm 0.02$ | $2.76\pm 0.70$ | $42.71\pm 0.010$
Swift/XRT | 57286 | $0.13\pm 0.64$ | $2.05\pm 0.01$ | $6.58\pm 0.67$ | $43.26\pm 0.003$ | $1.91\pm 0.03$ | $0.79\pm 0.71$ | $42.78\pm 0.012$
Swift/XRT | 57564 | $0.14\pm 0.62$ | $1.66\pm 0.03$ | $4.51\pm 0.64$ | $43.24\pm 0.008$ | $2.45\pm 0.04$ | $4.07\pm 0.72$ | $42.95\pm 0.011$
Swift/XRT | 57991 | $0.21\pm 0.61$ | $1.35\pm 0.03$ | $2.21\pm 0.63$ | $43.09\pm 0.016$ | $2.43\pm 0.03$ | $8.06\pm 0.78$ | $43.00\pm 0.007$
Swift/XRT | 58359 | $0.17\pm 0.69$ | $1.81\pm 0.03$ | $10.1\pm 0.67$ | $43.53\pm 0.006$ | $3.23\pm 0.03$ | $4.23\pm 0.71$ | $42.89\pm 0.013$
Swift/XRT | 58676 | $0.41\pm 0.65$ | $1.73\pm 0.03$ | $7.05\pm 0.61$ | $43.41\pm 0.007$ | $3.17\pm 0.04$ | $7.01\pm 0.72$ | $43.11\pm 0.009$
Swift/XRT | 59087 | $0.08\pm 0.67$ | $1.34\pm 0.02$ | $10.8\pm 0.64$ | $43.42\pm 0.003$ | $2.78\pm 0.04$ | $0.46\pm 0.70$ | $42.77\pm 0.021$
Ton S180
Instrument | MJD | $N_{\rm H}$ | $\Gamma^{\rm PC}$ | $Norm^{PC}$ | $\log(L^{\rm PC})$ | $\Gamma^{\rm SE}$ | $Norm^{SE}$ | $\log(L^{\rm SE})$
|  | $(10^{20}~{}cm^{-2})$ |  | $(10^{-3})$ | (erg/s) |  | $(10^{-3})$ | (erg/s)
XMM-Newton | 51892 | $2.88\pm 0.51$ | $2.10\pm 0.03$ | $1.96\pm 0.58$ | $43.91\pm 0.002$ | $4.05\pm 0.03$ | $2.64\pm 0.53$ | $43.95\pm 0.001$
XMM-Newton | 52455 | $1.13\pm 0.58$ | $1.93\pm 0.03$ | $1.02\pm 0.55$ | $43.67\pm 0.004$ | $2.43\pm 0.03$ | $2.43\pm 0.57$ | $43.75\pm 0.001$
XMM-Newton | 57206 | $0.83\pm 0.57$ | $2.06\pm 0.02$ | $1.52\pm 0.51$ | $43.81\pm 0.002$ | $3.98\pm 0.02$ | $1.68\pm 0.51$ | $43.74\pm 0.002$
XMM-Newton | 57552 | $0.70\pm 0.55$ | $2.13\pm 0.03$ | $0.99\pm 0.50$ | $43.60\pm 0.003$ | $3.56\pm 0.03$ | $1.06\pm 0.52$ | $43.51\pm 0.002$
Swift/XRT | 57907 | $2.07\pm 0.52$ | $1.14\pm 0.09$ | $0.50\pm 0.62$ | $43.78\pm 0.057$ | $3.44\pm 0.07$ | $2.80\pm 0.68$ | $43.92\pm 0.018$
Swift/XRT | 59339 | $2.95\pm 0.54$ | $1.15\pm 0.09$ | $0.34\pm 0.64$ | $43.82\pm 0.057$ | $3.13\pm 0.08$ | $2.88\pm 0.65$ | $43.93\pm 0.014$
UGC 6728
Instrument | MJD | $N_{\rm H}$ | $\Gamma^{\rm PC}$ | $Norm^{PC}$ | $\log(L^{\rm PC})$ | $\Gamma^{\rm SE}$ | $Norm^{SE}$ | $\log(L^{\rm SE})$
|  | $(10^{20}~{}cm^{-2})$ |  | $(10^{-3})$ | (erg/s) |  | $(10^{-3})$ | (erg/s)
XMM-Newton | 53789 | $0.81\pm 0.45$ | $1.39\pm 0.02$ | $1.01\pm 0.48$ | $41.72\pm 0.005$ | $3.01\pm 0.02$ | $1.47\pm 0.47$ | $41.52\pm 0.004$
Swift/XRT | 57579 | $0.89\pm 0.49$ | $1.14\pm 0.08$ | $0.53\pm 0.59$ | $41.42\pm 0.021$ | $2.82\pm 0.06$ | $1.03\pm 0.56$ | $38.74\pm 0.021$
Swift/XRT | 58039 | $3.74\pm 0.44$ | $1.66\pm 0.07$ | $3.50\pm 0.57$ | $42.04\pm 0.008$ | $4.57\pm 0.01$ | $0.20\pm 0.57$ | $41.78\pm 0.067$
Swift/XRT | 59097 | $2.84\pm 0.46$ | $1.25\pm 0.08$ | $0.61\pm 0.64$ | $41.80\pm 0.040$ | $3.73\pm 0.07$ | $2.17\pm 0.68$ | $41.76\pm 0.037$
Swift/XRT | 59414 | $1.02\pm 0.42$ | $1.23\pm 0.09$ | $0.86\pm 0.47$ | $41.72\pm 0.033$ | $3.06\pm 0.05$ | $2.61\pm 0.49$ | $41.71\pm 0.022$

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