# Constructing segments of quadratic length in $Spec(T_{n})$ through segments
of linear length

Artem Kravchuk<EMAIL_ADDRESS>Sobolev Institute of Mathematics,
Ak. Koptyug av. 4, Novosibirsk 630090, Russia  Novosibirsk State University,
Pirogova str. 2, Novosibirsk, 630090, Russia

###### Abstract

A _Transposition graph_ $T_{n}$ is defined as a Cayley graph over the
symmetric group $Sym_{n}$ generated by all transpositions. It is known that
the spectrum of $T_{n}$ consists of integers, but it is not known exactly how
these numbers are distributed. In this paper we prove that integers from the
segment $[-n,n]$ lie in the spectrum of $T_{n}$ for any $n\geqslant 31$. Using
this fact we also prove the main result of this paper that a segment of
quadratic length with respect to $n$ lies in the spectrum of $T_{n}$.

###### keywords:

Transposition graph; integral graph; spectrum;

###### MSC:

[2010] 05C25, 05E10, 05E15

††journal: Siberian Electronic Mathematical Reports

##  1 Introduction

A _Transposition graph_ $T_{n}$ is defined as a Cayley graph over the
symmetric group $Sym_{n}$ generated by all transpositions. The graph
$T_{n},n\geqslant 2$, is a connected bipartite $C_{n}^{2}$-regular graph [5].
The _spectrum_ of a graph is defined as a multiset of distinct eigenvalues of
its adjacency matrix together with their multiplicities. It was shown in [4,
Lemma 3] that $T_{n}$ is integral which means that its spectrum $Spec(T_{n})$
consists of integers. Later and independently in [8, Theorem 2], it was also
shown that $T_{n}$ is integral. Since the Transposition graph is bipartite,
its spectrum is symmetric about zero [1], i.e., if number $k$ lies in
$Spec(T_{n})$ then number $-k$ lies in $Spec(T_{n})$ too. The largest
eigenvalue of $T_{n}$ is equal to $C_{n}^{2}$ which implies that all
eigenvalues of $T_{n}$ lie in the interval $[-C_{n}^{2},C_{n}^{2}]$.

A nonincreasing sequence $p=(n_{1},\dots,n_{k})\vdash n$, $k\geqslant 1$, for
which $n=\sum\limits_{j=1}^{k}n_{j}$, is called a _partition_ of $n$. It is
follows from [4, eq. 3] that any partition $p\vdash n$ corresponds to an
eigenvalue $\lambda(p)\in Spec(T_{n})$ by the following expression:

$\lambda(p)=\sum_{j=1}^{k}\frac{n_{j}(n_{j}-2j+1)}{2}.$ (1)

The _conjugate of partition $p$_, denoted by $p^{\prime}$, is also a partition
of $n$ where its parts are the nonincreasing sequence
$p^{\prime}=(n_{1}^{\prime},n_{2}^{\prime},\ldots,n_{\kappa}^{\prime})$, where
$n_{j}^{\prime}=\sum\limits_{i|n_{i}\geqslant j}1$. This paper uses that
$n_{1}^{\prime}$ is equal to the length of $p$. It is follows from [2, Lemma
8] that

$\lambda(p)=-\lambda(p^{\prime}).$ (2)

Note that different partitions can correspond to the same eigenvalue. For
example, the partitions $(4,1,1)$ and $(3,3)$ correspond to the eigenvalue $3$
of $T_{6}$.

Thus, if one enumerates all partitions of number $n$ and substitutes them into
expression (1) one can obtain all eigenvalues of $T_{n}$. We write $f(n)\sim
g(n)$ if $\frac{f(n)}{g(n)}\rightarrow 1$ as $n\rightarrow\infty$. The number
of partitions $p(n)$ has the following asymptotics:
$p(n)\sim\frac{exp(\pi\sqrt{\frac{2}{3}}\sqrt{n-\frac{1}{24}})}{4n\sqrt{3}}$
[3], so it is no longer possible to enumerate all partitions for large $n$.

Moreover, looking at expression (1), it is even not possible to answer simple
questions about $Spec(T_{n})$:

  * •

is number $0$ an eigenvalue of $T_{n}$?

  * •

are all integers from the interval $[0,k],k\in\mathbb{N}$, lie in the spectrum
of $T_{n}$?

  * •

what is the asymptotics for the number of unique values in the spectrum of
$T_{n}$?

Ideally, one would like to be able to answer the question whether a given
integer number $k\in[-C_{n}^{2},C_{n}^{2}]$ lies in the spectrum of $T_{n}$.
The following result is known.

###### Theorem 1

[6, Theorem 2] For any integer $k\geqslant 0$, there exists $n(k)$ such that
for any $n\geqslant n(k)$ and any $m\in\\{0,\dots,k\\}$, $m\in Spec(T_{n})$.

In the proof of this theorem, $n(k)$ is chosen as $n(k)=10k+4$. Consequently,
all integers within the range $[-\frac{n-4}{10},\frac{n-4}{10}]$ are
encompassed within the spectrum of $T_{n}$. The following result extends the
result of Theorem 1.

###### Theorem 2

[7, Theorem 3] For any $n\geqslant 19$, all integers from the segment
$[-\frac{n-4}{2},\frac{n-4}{2}]$ lie in the spectrum of $T_{n}$.

The following result of this paper improves Theorem 2.

###### Theorem 3

For any $n\geqslant 31$, all integers from the segment $[-n,n]$ lie in the
spectrum of $T_{n}$.

Theorems 1, 2, 3 have in common that all integers in some neighborhood
$[-x,x]$ of zero lie in the spectrum of $T_{n}$. In the proof of these
theorems the segment $[0,x]$ is splitted into subsegments $S=\bigcup
s_{i},s_{i}=[\underline{x_{i}},\overline{x_{i}}]$ and individual integer
numbers $A=\\{a_{1},\dots,a_{k}\\}$ which may depend on the parity of $n$.
Moreover, $S$ and $A$ are chosen so that $S\bigcup A$ contains all integers
from the segment $[0,x]$. For each segment $s_{i}$, there is a family of
partitions such that all integers from the segment $s_{i}$ are covered by the
eigenvalues corresponding to the partitions from this family. Similar, for
each individual integer $a_{i}$, there is a partition $p_{i}$ such that
$\lambda(p_{i})=a_{i}$.

For example [6, Lemma 4], for any odd $n\geqslant 7$, the partition
$\bigl{(}\frac{n-2\lambda+1}{2},\lambda+2,2\times(\lambda-1),1\times\frac{n-4\lambda-1}{2}\bigr{)}$
corresponds to the eigenvalue $\lambda\in\mathbb{N}$, where
$\lambda\in[1,\frac{n-3}{4}]$. The notation $(n_{1},...,n_{j},2\times
t_{2},1\times t_{1})$ means that the number $2$ is repeated in the partition
$t_{2}$ times and the number $1$ is repeated $t_{1}$ times.

Thus, in the proof of Theorem 3, we obtain a set of partitions $P=\bigcup
p_{i}$ such that for each $k\in[0,n],k\in\bigcup\lambda(p_{i})$. Theorem 3
gives the following corollary, which is useful for the proof of Theorem 4.

###### Corollary 1

The first part of all partitions used to prove Theorem 3 does not exceed
$\frac{n+3}{2}$.

Note that Theorem 3 shows that all integers from the interval $[-n,n]$ lie in
the spectrum of $T_{n}$ and the length of this interval has asymptotics
$O(n)$.

The main result of this paper shows that all integers from the set
$[-y_{2},-y_{1}]\bigcup[y_{1},y_{2}]$ lie in the spectrum of $T_{n}$.
Moreover, the length of the segment $[y_{1},y_{2}]$ has $O(n^{2})$
asymptotics.

###### Theorem 4

For any $n\geqslant 48$, all integers from the segments $[-y_{2},-y_{1}]$ and
$[y_{1},y_{2}]$, lie in the spectrum of $T_{n}$, where
$y_{1}=C_{\lceil\frac{n}{3}\rceil+1}^{2}-2(\lfloor\frac{2n}{3}\rfloor-1)$ and
$y_{2}=C_{\lfloor\frac{2n+1}{3}\rfloor}^{2}$.

The paper is structured as follows. First, in Section 2 we prove Theorem 3.
The proof of Theorem 3 is based on several technical lemmas, which are proved
in Section 4. After proving Theorem 3, we use it and Corollary 1 to prove
Theorem 4 in Section 3.

##  2 Proof of Theorem 3

In what follows further the notation $S\in Spec(T_{n})$ where $S$ is a segment
$[\underline{x},\overline{x}]$ means that all integers from this segment are
eigenvalues of $T_{n}$. Similar, for finite set of numbers $A$ we say that
$A\in Spec(T_{n})$ if all elements of this set lie in the spectrum of $T_{n}$.

To prove the theorem, we use the approach described in [7]. We split the
segment $[0,n]$ into subsegments $S_{1},S_{2}$ and into two sets $A_{1}$,
$A_{2}$ of numbers depending on the parity of $n$ such that $S_{1}\cup
S_{2}\cup A_{1}\cup A_{2}$ contains all integers of the interval $[0,n]$. The
segments $S_{1},S_{2}$ and the sets $A_{1},A_{2}$ are shown in Table 1.

$n$ | $S_{1}$ | $A_{1}$ | $S_{2}$ |  $A_{2}$
---|---|---|---|---
odd | $[0,\frac{n-1}{2}]$ | $\\{\frac{n+1}{2},\frac{n+3}{2},\frac{n+5}{2}\\}$ | $[\frac{n+7}{2},n-7]$ |  $\\{n-6,n-5,\dots,n\\}$
even | $[0,\frac{n-4}{2}]$ | $\\{\frac{n-2}{2},\frac{n}{2},\frac{n+2}{2}\\}$ | $[\frac{n+4}{2},n-6]$
Table 1:  Splitting of segment $[0,n]$.

Next, we show that $S_{1}\in Spec(T_{n}),S_{2}\in Spec(T_{n})$, $A_{1}\in
Spec(T_{n})$ and $A_{2}\in Spec(T_{n})$. To show this, we prove the following
technical lemmas.

###### Lemma 1

$S_{1}\in Spec(T_{n})$ for any $n\geqslant 19$.

###### Corollary 2

The first part of all partitions together with their conjugates used to cover
$S_{1}$ does not exceed $\frac{n+1}{2}$.

###### Lemma 2

$S_{2}\in Spec(T_{n})$ for any $n\geqslant 20$.

###### Corollary 3

The first part in the partitions together with their conjugates used in the
proof of Lemma 1 does not exceed $\frac{n+2}{2}$.

###### Lemma 3

$A_{1}\in Spec(T_{n})$ for any $n\geqslant 31$.

###### Corollary 4

The first part in the partitions together with their conjugates used in the
proof of Lemma 3 does not exceed $\frac{n+2}{2}$.

###### Lemma 4

$A_{2}\in Spec(T_{n})$, for any $n\geqslant 19$.

###### Corollary 5

The first part in the partitions together with their conjugates used in the
proof of Lemma 4 does not exceed $\frac{n+3}{2}$.

The proof of these technical lemmas can be found in Section 4. After proving
the lemmas, it is directly deduced that the segment $[0,n]\in Spec(T_{n})$ for
any $n\geqslant 31$. Since $T_{n}$ is bipartite, we have $[-n,n]\in
Spec(T_{n})$. Since Corollaries 2-5 consider partitions together with their
conjugates, it follows from (2) that the first part of the partitions covered
by the segment $[-n,n]$ does not exceed $\frac{n+3}{2}$, which proves
Corollary 1.

##  3 Proof of Theorem 4

Let $p=(n_{1},n_{2},\dots,n_{k})=(n_{1},p_{1})$, where
$p_{1}=(n_{2},\dots,n_{k})$. It turns out that the eigenvalue $\lambda(p)$
corresponding to the partition $p$ can be compactly expressed through the
eigenvalue $\lambda(p_{1})$ corresponding to the partition $p_{1}$ and the
number $n_{1}$ using the following lemma.

###### Lemma 5

$\lambda(p)=C_{n_{1}}^{2}+\lambda(p_{1})-(n-n_{1}).$ (3)

Proof. We prove the lemma by a direct substitution of partition
$p=(n_{1},n_{2},\dots,n_{k})$ into the expression (1).

$\lambda(p)=\frac{1}{2}\sum_{j=1}^{k}n_{j}(n_{j}-2j+1)=\frac{n_{1}(n_{1}-1)}{2}+\frac{1}{2}\sum_{j=2}^{k}n_{j}(n_{j}-2j+1)=$
$=C_{n_{1}}^{2}+\frac{1}{2}\sum_{j=2}^{k}n_{j}(n_{j}-2(j-1)+1-2)=$
$=C_{n_{1}}^{2}+\sum_{j=2}^{k}n_{j}(n_{j}-2(j-1)+1)-\sum_{j=2}^{k}n_{j}=$
$=C_{n_{1}}^{2}+\lambda(p_{1})-(n-n_{1}).$

$\square$

Let us consider the expression (3). By varying $p_{1}$, one can obtain all
possible eigenvalues for the partitions that have $n_{1}$ in the first part.
The following conditions are imposed on $p_{1}$:

  1. 1.

$p_{1}\vdash n-n_{1}$;

  2. 2.

the first part of $p_{1}$ must not exceed $n_{1}$.

Let us denote the set of partitions that satisfy these two conditions by
$\mathbb{P}_{n_{1}}$ and the set of eigenvalues that correspond to these
partitions by $\bm{\lambda_{n_{1}}}$. If it is proved that all integers from
the interval $[-(n-n_{1}),n-n_{1}]$ lie in the $\bm{\lambda_{n_{1}}}$, then it
follows directly from (3) that $[C_{n_{1}}^{2}-2(n-n_{1}),C_{n_{1}}^{2}]\in
Spec(T_{n})$. Note that when $n_{1}\leqslant\frac{2n+1}{3}$ then
$2(n-n_{1})\geqslant n_{1}-1=C_{n_{1}}^{2}-C_{n_{1}-1}^{2}$. Thus, when
$n_{1}\leqslant\frac{2n+1}{3}$, then
$[C_{n_{1}}^{2}-2(n-n_{1}),C_{n_{1}}^{2}]\in Spec(T_{n})$ implies
$[C_{n_{1}-1}^{2},C_{n_{1}}^{2}]\in Spec(T_{n})$.

Consider the segment $l=[k_{1},k_{2}]$, where $k_{1}\in\mathbb{N}$ and
$k_{2}=\lfloor\frac{2n+1}{3}\rfloor$. If for each natural $n_{1}\in l$ it
holds that all integers from the interval $[-(n-n_{1}),n-n_{1}]$ lie in the
$\bm{\lambda_{n_{1}}}$, then it follows that
$[C_{k_{1}}^{2}-2(n-k_{1}),C_{k_{2}}^{2}]\in Spec(T_{n})$.

By Theorem 3, we have that if $n-n_{1}\geqslant 31$, then
$[-(n-n_{1}),n-n_{1}]\in Spec(T_{n-n_{1}})$. Moreover, by Corollary 1, to
cover the segment $[-(n-n_{1}),n-n_{1}]$ we are able to choose such partitions
that their first part does not exceed $\frac{n-n_{1}+3}{2}$. Inequality
$n_{1}\geqslant\frac{n-n_{1}+3}{2}$ can be rewritten as
$n_{1}\geqslant\frac{n}{3}+1$. So, if $n_{1}\geqslant\frac{n}{3}+1$ and
$n-n_{1}\geqslant 31$, then $[-(n-n_{1}),(n-n_{1})]\in\bm{\lambda_{n_{1}}}$.

Thus, for any natural $n_{1}\in[\frac{n}{3}+1,\frac{2n+1}{3}]$ and
$n-n_{1}\geqslant 31$ we have that
$[C_{n_{1}-1}^{2}-2(n-n_{1}),C_{n_{1}}^{2}]\in Spec(T_{n})$. Therefore,
$[C_{\lceil\frac{n}{3}\rceil+1}^{2}-2(n-\lceil\frac{n}{3}\rceil-1),C_{\lfloor\frac{2n+1}{3}\rfloor}^{2}]\in
Spec(T_{n})$ when $n-n_{1}\geqslant 31$. Note that
$n-\lceil\frac{n}{3}\rceil-1=\lfloor\frac{2n}{3}\rfloor-1$, hence
$[C_{\lceil\frac{n}{3}\rceil+1}^{2}-2(\lfloor\frac{2n}{3}\rfloor-1),C_{\lfloor\frac{2n+1}{3}\rfloor}^{2}]\in
Spec(T_{n})$. Moreover,
$[-C_{\lfloor\frac{2n+1}{3}\rfloor}^{2},-([C_{\lceil\frac{n}{3}\rceil+1}^{2}-2(\lfloor\frac{2n}{3}\rfloor-1))]\in
Spec(T_{n})$ too, since $T_{n}$ is bipartite. The inequalities
$n-n_{1}\geqslant 31$ and $n_{1}\geqslant\frac{n}{3}+1$ imply that $n\geqslant
48$, which completes the proof.

$\square$

##  4 Proof of technical lemmas

Let the partition $p$ is given by $p=(p_{l},2\times t_{2},1\times t_{1})$,
where $p_{l}$ is a sequence of length $l$. For example, a partition
$(5,4,4,2,2,2,1,1)$ can be represented as $(p_{3},2\times 3,1\times 2)$, where
$p_{3}=(5,4,4)$. Note that for a single partition the representation in this
form is not the only one. For example, we can represent the partition
$(5,4,4,2,2,2,1,1)$ as $(p_{4},2\times 2,1\times 2)$, where $p_{4}=(5,4,4,2)$.
Therefore, in what follows we emphasize how $p_{l}$ and $t_{1},t_{2}$ are
chosen.

###### Lemma 6

If $p=(p_{l},2\times t_{2},1\times t_{1})$, then

$\lambda(p)=\lambda(p_{l})-f(t_{1},t_{2},l)=\lambda(p_{l})-\underbrace{(t_{2}-1)^{2}-\frac{1}{2}(t_{1}-\frac{1}{2})^{2}-t_{1}t_{2}-l(2t_{2}+t_{1})+\frac{9}{8}}_{f(t_{1},t_{2},l)}.$

Proof. By (1) we have:

$\lambda(p)=\lambda(p_{l})+\frac{1}{2}\sum\limits_{j=l+1}^{l+t_{2}}2(3-2j)+\frac{1}{2}\sum\limits_{l+t_{2}+1}^{l+t_{1}+t_{2}}(2-2j)=$
$=\lambda(p_{l})+3t_{2}+t_{1}-2\sum\limits_{j=l+1}^{l+t_{2}}j-\sum\limits_{l+t_{2}+1}^{l+t_{1}+t_{2}}j=$

$=\lambda(p_{l})+3t_{2}+t_{1}-(2l+t_{2}+1)t_{2}-\frac{2l+2t_{2}+t_{1}+1}{2}t_{1}=$

$=\lambda(p_{l})-(t_{2}-1)^{2}-\frac{1}{2}(t_{1}-\frac{1}{2})^{2}-t_{1}t_{2}-l(2t_{2}+t_{1})+\frac{9}{8}.$

$\square$

###### Corollary 6

If $p=(p_{l},1\times t_{1})$, then

$\lambda(p)=\lambda(p_{l})-f(l,t_{1})=\lambda(p_{l})-\underbrace{t_{1}(\frac{t_{1}-1}{2}+l)}_{f(t_{1},l)}.$
(4)

Proof of Lemma 1. The proof is straightforward from the technical lemmas of
[7]. For the reader’s convenience, we have presented the summarization of
these lemmas in Table 4.

Proof of Corollary 2.

The proof if straightforward from Table 4. The largest value of the first part
is $\frac{n+1}{2}$ and is obtained on the partition
$(\frac{n+1}{2},1\times\frac{n-1}{2})$. The largest value among the conjugate
partitions is $\frac{n+1}{2}$ and is achieved on the partition conjugate to
$(\frac{n+1}{2},1\times\frac{n-1}{2})$. Note that the partition
$(\frac{n+1}{2},1\times\frac{n-1}{2})$ is self-conjugate and it follows from
(2) that if the partition $p$ is self-conjugate, then $\lambda(p)=0$.

Proof of Lemma 2. We prove this lemma by considering four cases, for each
parity of $n$ and $\lambda$. For each parity of $n$, we choose two families of
partitions $p_{\lambda}$ that depend on the parity of $\lambda$. Then we show
that eigenvalues corresponding to the chosen families of partitions cover all
integers from the segment $S_{2}$ which have the same parity as $\lambda$.

Case 1, $n$ is odd, $\lambda$ is odd:

$p_{\lambda}=\bigl{(}\underbrace{\frac{\lambda}{2}+\frac{3}{2},\frac{n+2}{2}-\frac{\lambda}{2},3}_{p_{l},l=3},2\times\underbrace{(\frac{n-4}{2}-\frac{\lambda}{2})}_{t_{2}},1\times\underbrace{(\lambda-\frac{n+3}{2})}_{t_{1}}\bigr{)}$.

By (1) we have:

$\lambda(p_{l})=\frac{1}{8}n^{2}-\frac{1}{4}n\lambda-\frac{1}{4}n+\frac{1}{4}\lambda^{2}+\frac{3}{4}\lambda-\frac{29}{8}.$

In addition, by (6) it follows that:

$f(t_{1},t_{2},l)=\frac{1}{8}n^{2}-\frac{1}{4}n\lambda-\frac{1}{4}n+\frac{1}{4}\lambda^{2}-\frac{1}{4}\lambda-\frac{29}{8}.$

Finally, using (6) we have
$\lambda(p)=\lambda(p_{l})+f(t_{1},t_{2},l)=\lambda$. Partition $p$ holds for
any odd integer $\lambda\in[\frac{n+3}{2},n-4]$. Note that the interval
$[\frac{n+3}{2},n-4]$ makes sense for any $n\geqslant 11$. Also notice that
segment $[\frac{n+7}{2},n-7]$ lies inside segment $[\frac{n+3}{2},n-4]$.

Case 2, $n$ is odd, $\lambda$ is even:

$p_{\lambda}=\bigl{(}\underbrace{\frac{\lambda}{2},\frac{n+3}{2}-\frac{\lambda}{2},5}_{p_{l},l=3},2\times\underbrace{(\frac{n-3}{2}-\frac{\lambda}{2})}_{t_{2}},1\times\underbrace{(\lambda-\frac{n+7}{2})}_{t_{1}}\bigr{)}$.

For $\lambda(p_{l})$ and $f(t_{1},t_{2},l)$ we have the following expressions:

$\lambda(p_{l})=\frac{1}{8}n^{2}-\frac{1}{4}n\lambda+\frac{1}{4}\lambda^{2}-\frac{1}{4}\lambda-\frac{9}{8};$

$f(t_{1},t_{2},l)=\frac{1}{8}n^{2}-\frac{1}{4}n\lambda+\frac{1}{4}\lambda^{2}-\frac{5}{4}\lambda-\frac{9}{8}.$

Finally, $\lambda(p)=\lambda(p_{l})+f(t_{1},t_{2},l)=\lambda$. Note that $p$
holds when even $\lambda\in[\frac{n+7}{2},n-7]$, and $[\frac{n+7}{2},n-7]$
holds for any $n\geqslant 21$.

Case 3, $n$ is even, $\lambda$ is odd:

$p_{\lambda}=\bigl{(}\underbrace{\frac{\lambda}{2}+\frac{3}{2},\frac{n+3}{2}-\frac{\lambda}{2}}_{p_{l},l=2},2\times\underbrace{(\frac{n-1}{2}-\frac{\lambda}{2})}_{t_{2}},1\times\underbrace{(\lambda-\frac{n+4}{2})}_{t_{1}}\bigr{)}$.

The following holds:

$\lambda(p_{l})=\frac{1}{8}n^{2}-\frac{1}{4}n\lambda+\frac{1}{4}\lambda^{2}+\frac{1}{2}\lambda-\frac{3}{4};$

$f(t_{1},t_{2},l)=\frac{1}{8}n^{2}-\frac{1}{4}n\lambda+\frac{1}{4}\lambda^{2}-\frac{1}{2}\lambda-\frac{3}{4}.$

Finally, $\lambda(p)=\lambda(p_{l})+f(t_{1},t_{2},l)=\lambda$. The following
constraints are imposed on $\lambda$ and $n$, so that $p$ holds: odd
$\lambda\in[\frac{n+4}{2},n-1]$, $n\geqslant 6$. Also notice that segment
$[\frac{n+4}{2},n-6]$ lies inside segment $[\frac{n+4}{2},n-1]$.

Case 4, $n$ is even, $\lambda$ is even:

$p_{\lambda}=\bigl{(}\underbrace{\frac{\lambda}{2}+1,\frac{n+2}{2}-\frac{\lambda}{2},4}_{p_{l},l=3},2\times\underbrace{(\frac{n-4}{2}-\frac{\lambda}{2})}_{t_{2}},1\times\underbrace{(\lambda-\frac{n+4}{2})}_{t_{1}}\bigr{)}.$

The following holds:

$\lambda(p_{l})=\frac{1}{8}n^{2}-\frac{1}{4}n\lambda-\frac{1}{4}n+\frac{1}{4}\lambda^{2}+\frac{1}{2}\lambda-3;$

$f(t_{1},t_{2},l)=\frac{1}{8}n^{2}-\frac{1}{4}n\lambda-\frac{1}{4}n+\frac{1}{4}\lambda^{2}-\frac{1}{2}\lambda-3.$

Note that $p$ holds for even $\lambda\in[\frac{n+4}{2},n-6]$ and for any
$n\geqslant 16$.

At the end of the proof, we would like to mention that for any $n\geqslant 20$
the restrictions for all four cases hold.

Proof of Corollary 3. We need to consider a family of partitions in each of
the four cases from Lemma 2. For each family, we have found an interval in
which $\lambda$ can lie. Let us substitute the maximum value from this
interval into the partitions and choose the maximum value at the first
position in these four cases. The maximum value is reached in Case 3 and is
equal to $\frac{n+2}{2}$. Moreover, the maximum value of the first part for
the conjugate partition is reached in Case 3 too and equal to $\frac{n-2}{2}$.

Proof of Lemma 3. For each $a\in A_{1}$ we give a particular partition
$p\vdash n$ such that $\lambda(p)=a$. All the given partitions have the form
$(p_{l},1\times l)$, so we use (4) to prove that $\lambda(p)=a$. For example,
for $\frac{n+1}{2}$ in the case of odd $n$, we take the partition
$p=\bigl{(}\underbrace{\frac{n-11}{2},7,4,3,3}_{p_{l},l=5},1\times\underbrace{(\frac{n-23}{2})}_{t_{1}}\bigr{)}\vdash
n$. For this partition we have:

$\lambda(p_{l})=\frac{1}{2}\bigl{(}\frac{n-11}{2}\cdot\frac{n-13}{2}+28-4-12-18\bigr{)}=\frac{1}{8}n^{2}-3n+\frac{119}{8},$

$f(t_{1},l)=\frac{n-23}{2}\bigl{(}\frac{n-25}{4}+5\bigr{)}=\frac{n^{2}}{8}-\frac{7}{2}n+\frac{115}{8}.$

Then, by (4) we get $\lambda(p)=\lambda(p_{l})-f(t_{1},l)=\frac{n+1}{2}$. Note
that $p$ holds for any $n\geqslant 25$. The given
$\lambda(p_{l}),f(t_{1},l),\lambda(p)$ together with the constraints on $n$
for the eigenvalue $\frac{n+1}{2}\in A_{1}$ can be found in the first row of
Table 2.

The partitions $p\vdash n$ along with $\lambda(p_{l})$ and $f(t_{1},l)$ for
the remaining numbers from $A_{1}$ are summarized in Table 2. In addition,
Table 2 shows the limitations under which the partition $p$ holds. Note that
for $n=31$ all limitations on $n$ listed in Table 2 are satisfied. $\square$

$p$ | $\lambda(p_{l})$ | $f(t_{1},l)$ | $\lambda(p)$ | limitations
---|---|---|---|---
|
$\bigl{(}\underbrace{\frac{n-11}{2},7,4,3,3}_{p_{l},l=5},1\times\underbrace{(\frac{n-23}{2})}_{t_{1}}\bigr{)}$
---
$\bigl{(}\underbrace{\frac{n-1}{2},4}_{p_{l},l=2},1\times\underbrace{(\frac{n-7}{2})}_{t_{1}}\bigr{)}$
$\bigl{(}\underbrace{\frac{n-3}{2},5,2}_{p_{l},l=3},1\times\underbrace{(\frac{n-11}{2})}_{t_{1}}\bigr{)}$
| $\frac{1}{8}n^{2}-3n+\frac{119}{8}$
---
$\frac{1}{8}n^{2}-\frac{1}{2}n+\frac{19}{8}$
$\frac{1}{8}n^{2}-n+\frac{31}{8}$
| $\frac{1}{8}n^{2}-\frac{7}{2}n+\frac{115}{8}$
---
$\frac{1}{8}n^{2}-n+\frac{7}{8}$
$\frac{1}{8}n^{2}-\frac{3}{2}n+\frac{11}{8}$
| $\frac{n+1}{2}$
---
$\frac{n+3}{2}$
$\frac{n+5}{2}$
| $odd\ n\geqslant 25$
---
$odd\ n\geqslant 9$
$odd\ n\geqslant 13$
|
$\bigl{(}\underbrace{\frac{n-16}{2},8,5,4,3,3}_{p_{l},l=6},1\times\underbrace{(\frac{n-30}{2})}_{t_{1}}\bigr{)}$
---
$\bigl{(}\underbrace{\frac{n+2}{2}}_{p_{l},l=1},1\times\underbrace{(\frac{n-2}{2})}_{t_{1}})$
$(\underbrace{\frac{n-12}{2},8,3,3,3,2}_{p_{l},l=6},1\times\underbrace{(\frac{n-26}{2})}_{t_{1}}\bigr{)}$
| $\frac{1}{8}n^{2}-\frac{17}{4}n+29$
---
$\frac{1}{8}n^{2}+\frac{1}{4}n$
$\frac{1}{8}n^{2}-\frac{13}{4}n+14$
| $\frac{1}{8}n^{2}-\frac{19}{4}n+30$
---
$\frac{1}{8}n^{2}-\frac{1}{4}n$
$\frac{1}{8}n^{2}-\frac{15}{4}n+13$
| $\frac{n-2}{2}$
---
$\frac{n}{2}$
$\frac{n+2}{2}$
| $even\ n\geqslant 32$
---
$even\ n\geqslant 2$
$even\ n\geqslant 28$
Table 2:  Partitions $p\vdash n$ and calculation of $\lambda(p)$ for them for
$A_{1}$.

Proof of Corollary 4. The proof is straightforward from the first column of
Table 2. The maximum value of the first part among all partitions is
$\frac{n+2}{2}$ and is obtained on the partition
$\bigl{(}\frac{n+2}{2},1\times(\frac{n-2}{2})\bigr{)}$. For conjugate
partitions the maximum is obtained on the partition which is conjugate to the
partition $\bigl{(}\frac{n+2}{2},1\times(\frac{n-2}{2})\bigr{)}$ and is equal
to $\frac{n}{2}$.

Proof of Lemma 4. We prove this lemma in a similar manner as Lemma 3. For
every $a\in A_{2}$ and every parity of $n$, we give a partition $p\vdash n$
such that $\lambda(p)=a$. For example, if $a=n$ and $n$ is odd we take the
partition
$p=\bigl{(}\underbrace{\frac{n+3}{2}}_{p_{l},l=1},1\times\underbrace{(\frac{n-3}{2})}_{t_{1}}\bigr{)}\vdash
n$, for which we have:

$\lambda(p_{l})=\frac{1}{2}\cdot\frac{n+3}{2}\cdot\frac{n+1}{2}=\frac{1}{8}n^{2}+\frac{1}{2}n+\frac{3}{8},$

$f(t_{1},l)=\frac{n-3}{2}\cdot\frac{n-1}{4}=\frac{1}{8}n^{2}-\frac{1}{2}n+\frac{3}{8}.$

By (4) we have that $\lambda(p)=\lambda(p_{l})-f(t_{1},l)=n$. Note that $p$
holds for any $n\geqslant 3$. The given $\lambda(p_{l}),f(t_{1},l),\lambda(p)$
together with the constraints on $n$ for the odd $n\in A_{2}$ are located in
the first row of Table 3.

The partitions $p\vdash n$ along with $\lambda(p_{l})$ and $f(t_{1},l)$ for
the remaining numbers from $A_{2}$ are summarized in Table 3. This table is
divided into cells for each $a\in A_{2}$. The top row of each cell shows the
partition for odd $n$, and the bottom row shows the partition for even $n$. In
addition, Table 3 shows the limitations under which the partition $p$ holds.
Note that for $n=19$ all limitations on $n$ listed in the table are satisfied.
$\square$

$p$ | $\lambda(p_{l})$ | $f(t_{1},l)$ | $\lambda(p)$ |  $n$ limitations
---|---|---|---|---
|
$\bigl{(}\underbrace{\frac{n+3}{2}}_{p_{l},l=1},1\times\underbrace{(\frac{n-3}{2})}_{t_{1}}\bigr{)}$
---
$\bigl{(}\underbrace{\frac{n}{2},4}_{p_{l},l=2},1\times\underbrace{(\frac{n}{2}-4)}_{t_{1}}\bigr{)}$
| $\frac{1}{8}n^{2}+\frac{1}{2}n+\frac{3}{8}$
---
$\frac{1}{8}n^{2}-\frac{1}{4}n+2$
| $\frac{1}{8}n^{2}-\frac{1}{2}n+\frac{3}{8}$
---
$\frac{1}{8}n^{2}-\frac{5}{4}n+2$
| $n$
---
| $n\geqslant 3$
---
$n\geqslant 8$
|
$\bigl{(}\underbrace{\frac{n+1}{2},3}_{p_{l},l=2},1\times\underbrace{(\frac{n-7}{2})}_{t_{1}}\bigr{)}$
---
$\bigl{(}\underbrace{\frac{n+2}{2},2}_{p_{l},l=2},1\times\underbrace{(\frac{n}{2}-3)}_{t_{1}}\bigr{)}$
| $\frac{1}{8}n^{2}-\frac{1}{8}$
---
$\frac{1}{8}n^{2}+\frac{n}{4}-1$
| $\frac{1}{8}n^{2}-n+\frac{7}{8}$
---
$\frac{1}{8}n^{2}-\frac{3}{4}n$
| $n-1$
---
| $n\geqslant 7$
---
$n\geqslant 6$
|
$\bigl{(}\underbrace{\frac{n-1}{2},4,2}_{p_{l},l=3},1\times\underbrace{(\frac{n-11}{2})}_{t_{1}}\bigr{)}$
---
$\bigl{(}\underbrace{\frac{n-8}{2},6,5,3}_{p_{l},l=4},1\times\underbrace{(\frac{n}{2}-10)}_{t_{1}}\bigr{)}$
| $\frac{1}{8}n^{2}-\frac{n}{2}-\frac{5}{8}$
---
$\frac{1}{8}n^{2}-\frac{9}{4}n+13$
| $\frac{1}{8}n^{2}-\frac{3}{2}n+\frac{11}{8}$
---
$\frac{1}{8}n^{2}-\frac{13}{4}n+15$
| $n-2$
---
| $n\geqslant 11$
---
$n\geqslant 20$
|
$\bigl{(}\underbrace{\frac{n+1}{2},2,2}_{p_{l},l=3},1\times\underbrace{(\frac{n-9}{2})}_{t_{1}}\bigr{)}$
---
$\bigl{(}\underbrace{\frac{n}{2},3,2}_{p_{l},l=3},1\times\underbrace{(\frac{n-10}{2})}_{t_{1}}\bigr{)}$
| $\frac{1}{8}n^{2}-\frac{33}{8}$
---
$\frac{1}{8}n^{2}-\frac{1}{4}n-3$
| $\frac{1}{8}n^{2}-n-\frac{9}{8}$
---
$\frac{1}{8}n^{2}-\frac{5}{4}n$
| $n-3$
---
| $n\geqslant 9$
---
$n\geqslant 10$
|
$\bigl{(}\underbrace{\frac{n-1}{2},3,3}_{p_{l},l=3},1\times\underbrace{(\frac{n-11}{2})}_{t_{1}}\bigr{)}$
---
$\bigl{(}\underbrace{\frac{n-4}{2},5,3,2}_{p_{l},l=4},1\times\underbrace{(\frac{n-16}{2})}_{t_{1}}\bigr{)}$
| $\frac{1}{8}n^{2}-\frac{1}{2}n-\frac{21}{8}$
---
$\frac{1}{8}n^{2}-\frac{5}{4}n$
| $\frac{1}{8}n^{2}-\frac{3}{2}n+\frac{11}{8}$
---
$\frac{1}{8}n^{2}-\frac{9}{4}n+4$
| $n-4$
---
| $n\geqslant 11$
---
$n\geqslant 16$
|
$\bigl{(}\underbrace{\frac{n-7}{2},5,5,3}_{p_{l},l=4},1\times\underbrace{(\frac{n-19}{2})}_{t_{1}}\bigr{)}$
---
$\bigl{(}\underbrace{\frac{n-2}{2},4,2,2}_{p_{l},l=4},1\times\underbrace{(\frac{n-14}{2})}_{t_{1}}\bigr{)}$
| $\frac{1}{8}n^{2}-2n+\frac{55}{8}$
---
$\frac{1}{8}n^{2}-\frac{3}{4}n-5$
| $\frac{1}{8}n^{2}-3n+\frac{95}{8}$
---
$\frac{1}{8}n^{2}-\frac{7}{4}n$
| $n-5$
---
| $n\geqslant 19$
---
$n\geqslant 14$
|
$\bigl{(}\underbrace{\frac{n-1}{2},3,2,2}_{p_{l},l=4},1\times\underbrace{(\frac{n-13}{2})}_{t_{1}}\bigr{)}$
---
$\bigl{(}\underbrace{\frac{n}{2},2,2,2}_{p_{l},l=4},1\times\underbrace{(\frac{n-12}{2})}_{t_{1}}\bigr{)}$
| $\frac{1}{8}n^{2}-\frac{1}{2}n-\frac{61}{8}$
---
$\frac{1}{8}n^{2}-\frac{1}{4}n-9$
| $\frac{1}{8}n^{2}-\frac{3}{2}n-\frac{13}{8}$
---
$\frac{1}{8}n^{2}-\frac{5}{4}n-3$
| $n-6$
---
| $n\geqslant 13$
---
$n\geqslant 12$
Table 3:  Partitions $p\vdash n$ and calculation of $\lambda(p)$ for them for
$A_{2}$.

Proof of Corollary 5. The proof is straightforward from the first column of
Table 3. The partition with maximum first part is
$(\frac{n+3}{2},1\times(\frac{n-3}{2}))$. The maximum of the first part among
conjugate partitions is achieved by a partition conjugate to
$\bigl{(}\frac{n+3}{2},1\times(\frac{n-3}{2})\bigr{)}$ and is equal to
$\frac{n-1}{2}$.

##  5 Discussions and further research

In this paper it is shown that for any $n\geqslant 48$, $[y_{1},y_{2}]\in
Spec(T_{n})$ and $[-y_{2},y_{1}]\in Spec(T_{n})$, where
$y_{1}=C_{\lceil\frac{n}{3}\rceil+1}^{2}-2(\lfloor\frac{2n}{3}\rfloor-1)$ and
$y_{2}=C_{\lfloor\frac{2n+1}{3}\rfloor}^{2}$. Note that both of these segments
have length with asymptotics $O(n^{2})$. In addition, it was shown that
$[-n,n]\in Spec(T_{n})$ for any $n\geqslant 31$.

Our conjecture is that the segment $[n+1,y_{1}]$ lies in $Spec(T_{n})$ too and
we plan to study this in future works. If this conjecture is true, it would
prove the following conjecture.

Conjecture.  There exists $n_{0}$ such that
$[-C_{\lfloor\frac{2n+1}{3}\rfloor}^{2},C_{\lfloor\frac{2n+1}{3}\rfloor}^{2}]\in
Spec(T_{n})$ for any $n\geqslant n_{0}$.

## Acknowledgements

The author was supported by the Mathematical Center in Akademgorodok, under
agreement No. 075-15-2022-281 with the Ministry of Science and High Education
of the Russian Federation.

Eigenvalues, $\lambda$ | Partitions | Limitations | Proof
---|---|---|---
$\ \ \ \ \ 0$ | $\bigl{(}\frac{n+1}{2},1\times\frac{n-1}{2}\bigr{)}$ |  $n$ is odd, $n\geqslant 1$ |  [6], Lemma 3
$\bigl{(}\frac{n}{2},2,1\times\frac{n-4}{2}\bigr{)}$ |  $n$ is even, $n\geqslant 4$
$[1,\frac{n-3}{4}]$ | $(\frac{n-2\lambda+1}{2},\lambda+2,2\times(\lambda-1),1\times\frac{n-4\lambda-1}{2})$ |  $n$ is odd, $n\geqslant 7$ |  [6], Lemma 5
$1$ | $\bigl{(}\frac{n-6}{2},4,4,2,1\times\frac{n-14}{2}\bigr{)}$ |  $n$ is even, $n\geqslant 14$ |  [6], Lemma 4
$[2,\frac{n-4}{4}]$ | $(\frac{n-2\lambda}{2},\lambda+2,3,2\times(\lambda-2),1\times\frac{n-4\lambda-2}{2})$ |  $n$ is even, $n\geqslant 12$ |  [7], Lemma 4
$[\frac{n+3}{4},\frac{n-1}{2}]$ | $\bigl{(}\lambda+1,\frac{n+3-2\lambda}{2},2\times(\frac{n-1}{2}-\lambda),1\times(\frac{4\lambda-n-3}{2})\bigr{)}$ |  $n$ is odd, $n\geqslant 5$ |  [7], Lemma 5
$[\frac{n+2}{4},\frac{n-4}{2}]$ | $\bigl{(}\lambda+1,\frac{n+2-2\lambda}{2},3,2\times(\frac{n-4}{2}-\lambda),1\times(\frac{4\lambda-n-2}{2})\bigr{)}$ |  $n$ is even, $n\geqslant 10$ |  [7], Lemma 6
$[\frac{n-3}{4}]+1$ | $\bigl{(}\frac{n}{4},\frac{n}{4},5,4,2\times(\frac{n-20}{4}),1\bigr{)}$ | $n\equiv 0\ (mod\ 4),n\geqslant 20$ |  [7], Lemma 7
$\bigl{(}\frac{n+3}{4},\frac{n+3}{4},4,2\times(\frac{n-13}{4}),1\bigr{)}$ | $n\equiv 1\ (mod\ 4),n\geqslant 13$ |  [7], Lemma 8
$\bigl{(}\frac{n+2}{4},\frac{n-2}{4},5,4,2\times(\frac{n-22}{4}),1,1\bigr{)}$ | $n\equiv 2\ (mod\ 4),n\geqslant 22$ |  [7], Lemma 9
$\bigl{(}\frac{n+1}{4},\frac{n+1}{4},5,3,2\times(\frac{n-19}{4}),1\bigr{)}$ | $n\equiv 3\ (mod\ 4),n\geqslant 19$ |  [7], Lemma 10
Table 4:  Summary of the technical lemmas from [7].

## References

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  * [2] P. Diaconis; M. Shahshahani, Generating a random permutation with random transpositions. _Z. Wahrsch. Verw. Gebiete_ , 57 (2) (1981) 159–179. https://doi.org/10.1007/BF00535487
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  * [8] E. V. Konstantinova, D. V. Lytkina, Integral Cayley graphs over finite groups, _Algebra Colloquium_ , 27(1) (2020) 131–136, https://doi.org/10.1142/S1005386720000115


# OGLE-2007-BLG-224L: Confirmation of Terrestrial Parallax

Yutong Shan  Institut für Astrophysik, Georg-August-Universität, Friedrich-
Hund-Platz 1, 37077 Göttingen, Germany  Jennifer C. Yee  Center for
Astrophysics $|$ Harvard & Smithsonian, 60 Garden St., Cambridge, MA 02138
Vanessa P. Bailey  Jet Propulsion Laboratory, 4800 Oak Grove Dr, M/S 321-123,
Pasadena, CA 91109  Laird M. Close  Steward Observatory, University of
Arizona, 933 North Cherry Avenue, Tucson, AZ 85721, USA  Phil M. Hinz
Department of Astronomy and Astrophysics, University of California, Santa
Cruz, 1156 High St, Santa Cruz, CA 95064, USA  Jared R. Males  Steward
Observatory, University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721,
USA  Katie M. Morzinski  Steward Observatory, University of Arizona, 933 North
Cherry Avenue, Tucson, AZ 85721, USA

###### Abstract

We present limits on the lens flux of OGLE-2007-BLG-224 based on MagAO imaging
taken seven years after the microlensing event. At the time of the
observations, the lens should have been separated from the microlensing source
by 292 mas. However, no new sources are detected with MagAO. We place an upper
limit on the lens flux of $H>20.57$. This measurement supports the conclusion
of Gould et al. (2009) that the lens in this event should be a brown dwarf.
This is the first test of a prediction based on the terrestrial microlens
parallax effect and the first AO confirmation of a sub-stellar/dark microlens.

gravitational lensing: micro – star: brown dwarfs

##  1 Introduction

Measurements of the terrestrial parallax effect in microlensing (Holz & Wald,
1996; Gould, 1997) should be extremely rare because the radius of the Earth is
much smaller than the typical size of the Einstein ring projected onto the
observer plane (1–10 au). This effect arises when the separation between two
observatories on the Earth is sufficient to produce a measurable difference in
the source magnification observed by each observatory. Incidentally, the same
conditions that lead to measurable terrestrial parallax effects are optimal
for measuring the finite source effect (Gould & Yee, 2013). From these two
effects one can measure the microlens parallax, $\pi_{\rm E}$, and the
Einstein radius, $\theta_{\rm E}$, respectively, and hence, derive the mass of
the lens and the distance to the system:

$M_{\rm L}=\frac{\theta_{\rm E}}{\kappa\pi_{\rm E}};\quad D_{\rm
L}=\left((\theta_{\rm E}\pi_{\rm E})\ \mathrm{au}+\frac{\mathrm{au}}{D_{\rm
S}}\right)^{-1}$ (1)

where $D_{\rm S}$ is the distance to the source (usually the mean distance to
the Galactic bulge) and $\kappa\equiv 8.14\ \mathrm{mas}\
\mathrm{M}_{\odot}^{-1}$. Gould & Yee (2013) estimated that only 0.1 mass
measurements from the combination of terrestrial parallax and finite source
effects should have been made to date (at the time that paper was written). By
contrast, two such measurements had been made.

One of these measurements was made for OGLE-2007-BLG-224 by Gould et al.
(2009)111The other was for OGLE-2008-BLG-279 (Yee et al., 2009).
OGLE-2007-BLG-224 was magnified by a factor of over 2500, and because of this
extreme magnification, both finite source effects and the terrestrial
microlens parallax effect were measured. The combination of the finite source
effect and the microlens parallax led to the conclusion that the lens was a
$0.056\pm 0.004M_{\odot}$ thick-disk brown dwarf at a distance of 525 pc with
the corresponding prediction that the lens should be very faint: $H=25.7$ mag.

In this paper, we present adaptive optics observations from MagAO that confirm
the lens is substellar or non-luminous.

##  2 Magellan Adaptive Optics Imaging

Figure 1: Left: MagAO imaging from April 2014 of the $8\farcs 8\times 8\farcs
8$ field surrounding OGLE-2007-BLG-224. The position of the microlensing
source is indicated by the arrow. The dashed squares mark the stars used for
constructing the empirical PSF. The solid rectangle marks the extent of the
substamp shown in Figure 2. Right: HST NICMOS F160W imaging of the same field
from June 2008.

###  2.1 Observations

OGLE-2007-BLG-224 was observed on the night of 20 April 2014 using the MagAO
instrument on the Magellan Clay Telescope (Morzinski et al., 2016). For these
observations, we used the Clio camera in “wide” mode, which gives a pixel
scale of 27.49 mas pix-1 and a $14^{\prime\prime}\times 28^{\prime\prime}$
field-of-view. The observations were taken using a four point dither pattern.
At each position, five 30s exposures were taken. The observations were made in
the $H$ band. A sky field was also observed earlier in the night.

###  2.2 Image Processing

The data were flat-fielded using a difference flat constructed from flats that
were taken at twilight on 21 April 2014. They were then sky subtracted using
the mean sky frame. Bad pixels were flagged using a custom bad pixel mask,
based on pixels with either large standard deviations in the flat frames
(“hot” pixels) or low counts (“dead” pixels). After sky subtraction, we found
that the median counts for each frame varied as a function of time (by
$\lesssim 1\%$ of the raw counts). We measured an offset for each frame and
subtracted it. Then, we re-sampled the images at a factor of 5 higher
resolution before shifting them and taking the median of the images. The final
image is downsampled from this median image to the original resolution.

The location of the target was measured from observations taken with ANDICAM
(DePoy et al., 2003) on the SMARTS 1.3m telescope in Chile, while the event
was highly magnified, i.e., on HJD${}^{\prime}=$HJD$-2,450,000=4233.6$. These
observations were taken in both $I$ and $H$ bands. Only one star in the
ANDICAM $H$-band data (other than the target) is bright enough to be detected
and also in the MagAO field. Thus, we matched the catalog for this frame to
the VVV catalog (Minniti et al., 2017), and then the VVV catalog to a catalog
of stars extracted from the MagAO field using Source Extractor (Bertin &
Arnouts, 1996) to determine the location of the target. We also checked the
microlens source position by matching the ANDICAM $I$-band catalog directly to
the MagAO field and found consistent results. The identified target matches
the description of the field given by Gould et al. (2009).

For the astrometric solution, we used the full $14^{\prime\prime}\times
28^{\prime\prime}$ image. However, the two chips of the Clio detector have
different photometric responses. Thus, we restricted our flux measurements to
the chip containing the science target.

###  2.3 Image Analysis

Because the region surrounding OGLE-2007-BLG-224 is relatively crowded
(Figures 1 and 2), we fit sources using an empirical PSF constructed from four
bright, relatively isolated, stars surrounding the target. The empirical PSF
is parameterized by the centroid position (X, Y) and a scaling factor (peak
counts). We calibrate the MagAO fluxes by cross-matching to the VVV catalog
(Minniti et al., 2017). All of the VVV sources are blends of multiple MagAO
stars. For each VVV star, we fit for all sources in a $2\farcs 0$ substamp
around the target. We then sum the fluxes for all MagAO stars within $0\farcs
7$ of the VVV source. In total, there were 8 calibration stars spanning a
magnitude range from $H_{\rm VVV}=$ 15.8 to 12.8. From these stars, we derived
the flux relation:

$H_{\rm VVV}=H_{\rm Clio}+6.813\pm 0.015$ (2)

where $H_{\rm Clio}$ is calculated from the peak counts in the PSF (i.e.,
$H_{\rm Clio}=18-2.5\log_{10}{f_{\rm peak}}$).

To estimate the uncertainties in the measured fluxes, we divided the 20
exposures into two subsets and co-added each subset into new composite images
1 and 2. Then we measured the fluxes of stars in each composite image and took
the difference of the two measurement,
$\Delta_{i}=(\mathrm{flux}_{i,1}-\mathrm{flux}_{i,2})$, for each star, $i$. In
the limit as the number of stars at a particular apparent magnitude goes to
infinity, the uncertainty in the flux in the original co-added image is

$\sigma_{\rm tot}\equiv\frac{\sqrt{\langle\Delta^{2}\rangle}}{2}$ (3)

for that magnitude. In a real image, there is a finite number of stars
spanning a finite set of discrete flux values. So to estimate
$\langle\Delta^{2}\rangle$, we assume that the uncertainty in the flux is a
smooth function of the peak flux. We fit a power-law to the measured
$\Delta_{i}^{2}$ as a function of $\mathrm{flux}_{i}$ from the original image.
Solving Equation 3 for $\sigma_{\rm tot}$ gives us the typical measurement
error as a function of flux.

##  3 HST Archival Data

The field of OGLE-2007-BLG-224 was observed by HST-NICMOS on 11 June 2007 and
8 June 2008 (PI: K. Sahu), respectively about one month and one year after the
microlensing event peak. To compare to our MagAO imaging, we retrieved the
F160W images taken in 2008 from the HST Legacy Archive. We identified the
brightest isolated star in the field to use as our $1\farcs\times 1\farcs$ PSF
template.

##  4 Non-Detection of the Lens

Figure 2: MagAO $2\farcs 0\times 1\farcs 3$ substamp showing the microlensing
source from OGLE-2007-BLG-224 (#1) and other nearby stars (#2–11). The large,
white circle surrounding the source shows a separation of $292\pm 14$ mas
(solid and dotted lines, respectively), which is the expected separation of
the lens from the source based on the magnitude of the proper motion alone.
The magenta circle is the expected position of the lens including the
microlens parallax measurement of the direction of the proper motion. Top:
MagAO from 2014, middle: residuals to the MagAO image after PSF fitting (the
colorbar ranges from $-20$ to $20$ counts), bottom: HST F160W imaging from
2008. The HST PSF has a ring feature corresponding to the first Airy ring.

Based on the analysis of Gould et al. (2009), we expect the lens to be a brown
dwarf and, therefore, faint and undetectable in the MagAO images. We can
predict the location of the lens based on the lens-source relative proper
motion measured by Gould et al. (2009):

$\displaystyle\mu_{\rm hel}$ $\displaystyle=$ $\displaystyle\tilde{v}_{\rm
hel}\left(\frac{\pi_{\rm rel}}{\mathrm{au}}\right)=42.06\pm 2\ \mathrm{mas\
yr}^{-1}$ (4) $\displaystyle\hat{\mu}_{\rm hel}$ $\displaystyle=$
$\displaystyle 61^{\circ}\pm 5^{\circ}\ \mathrm{S\ of\ W}.$ (5)

Thus, at the time of the MagAO observations the lens and microlensing source
are expected to have separated by $292\pm 14$ mas.

The expected location of the lens is shown in Figure 2. The magenta circle
shows the predicted location of the lens, including the measurement of the
direction of the proper motion from the terrestrial parallax. The white
annulus shows the constraints on the predicted lens location based on the
measurement of $\theta_{\rm E}$ alone. If the terrestrial parallax measurement
were in error, and the lens were a main sequence star, we would expect an
additional source to be detected somewhere along this annulus.

###  4.1 Fitting for Known Sources

Figure 3: Fluxes for stars measured in the MagAO image are well-matched to
fluxes measured from the 2008 HST image. In particular, no significant
deviation is seen for the stars closest to the microlensing source (red
circles, numbered as in Figure 2).  Table 1: Properties of Sources Near
OGLE-2007-BLG-224

Source ID | $X_{\rm centroid}$ | $Y_{\rm centroid}$ | $H_{\rm 2MASS}$ |  |  |
---|---|---|---|---|---|---
| (pixels) | (pixels) |  |  |  |
1 | 346.790 | 298.883 | 17.38$\pm$0.04 |  |  |
2 | 344.522 | 304.047 | 19.18$\pm$0.07 |  |  |
3 | 358.467 | 299.352 | 16.89$\pm$0.03 |  |  |
4 | 352.726 | 286.942 | 20.10$\pm$0.11 |  |  |
5 | 326.571 | 284.812 | 16.10$\pm$0.03 |  |  |
6 | 317.868 | 276.211 | 20.28$\pm$0.12 |  |  |
7 | 317.360 | 293.629 | 21.29$\pm$0.20 |  |  |
8 | 352.372 | 275.179 | 20.07$\pm$0.11 |  |  |
9 | 337.679 | 322.640 | 18.59$\pm$0.06 |  |  |
10 | 330.436 | 312.293 | 21.62$\pm$0.24 |  |  |
11 | 322.803 | 317.831 | 21.37$\pm$0.21 |  |  |

Note: Sources are numbered as in Figure 2

We begin by fitting for sources in a rectangular ($2\farcs 0\times 1\farcs 3$)
substamp (Figure 2) containing the source star of the event and its immediate
neighbors. These fits use the empirical PSF described in Section 2.3 plus a
constant background level for the image. Sources were recovered iteratively by
examining the residuals to the fitted image and comparing them to the HST
NICMOS images of the field. In total, 11 sources were recovered in the
($2\farcs 0\times 1\farcs 3$) substamp. All of them can be cross-matched to
stars in the HST NICMOS imaging. Their properties are given in Table 1. The
residuals to the eleven-source fit are shown in Figure 2. There is no
indication of an additional source within the white annulus.

The one exception to this could be if the lens were very near to source #3,
which happens to lie at the expected lens-source separation. In that case,
light from the lens might be absorbed into the fits for source #3. To check
for this possibility we compare the fluxes of the stars to the fluxes measured
from the HST image from 2008 (Figure 3). The fluxes for all the sources are
consistent between MagAO and HST, with no significant excesses. Therefore, we
conclude that source #3 is not hiding significant light from the lens.

Finally, from the MagAO image, we measure $H=17.38\pm 0.04$ for the source at
the position of OGLE-2007-BLG-224 (source #1). This is slightly brighter than
the microlens source flux as reported in Gould et al. (2009) ($H_{S}=17.50\pm
0.02$), which is marginally inconsistent at $2.7\sigma$. The excess would
correspond to an $H=19.8$ star. However, the measured source flux is also
consistent with the flux measured for the same source in the HST imaging.
Gould et al. (2009) state that the HST source is consistent with zero
additional flux aside from the microlens source. Even if the excess is real,
it is in the wrong place to be the lens, although it could be a companion to
the microlens source.

###  4.2 Search for Additional Sources

To confirm that there are no additional sources in the field, we conduct a
blind search. We seed fits for a twelfth source at every pixel over a $1\farcs
35\times 1\farcs 35$ square surrounding OGLE-2007-BLG-224S. The peak flux for
the PSF is initialized at 50 counts, and then this source is fit freely
simultaneously with the eleven known sources. We consider a source detected if
the final fit is inside the search grid, within 20 pixels of the starting
location, and has a fitted peak flux of at least 10 counts. We tested our
method for eleven-star fits to confirm that the two faintest sources (10 and
11) are recovered.

There are two classes of “detections” that survive these cuts. First, there
are “detections” very close to the centroids of known sources in the field. In
these cases, the twelfth source partially supplants the known source; the sum
of their fluxes remains the same as for the known source alone (in particular,
this is true for source #3, i.e., there is no evidence for an additional
source “hidden” in the wings of this star). Hence, we apply an additional cut
that the new detection should be at least 1 FWHM from a previously known
source to eliminate this class of “detections.” Second, there are “detections”
of a twelfth source along the bottom edge of the substamp. There are
unexplained residuals in this region at the level of $\sim 20$ counts,
possibly from unresolved faint sources, associated with the wings of bright
sources outside the substamp, or simply spurious. However, because this string
of “detections” are also separated from the source by almost twice the
expected distance travelled by the lens, even if they are real, they are
unlikely to be the lens.

Thus, this blind search did not reveal any new sources that are plausible
candidates for the lens of OGLE-2007-BLG-224.

###  4.3 Limits on Additional Stars

To assess the limits on non-detections in the image, we inject fake sources
into the image and attempt to recover them. Fake sources are injected at every
pixel over the $1\farcs 35\times 1\farcs 35$ grid used for the blind search.
Then, fits are seeded on a $3\times 3$ ‘mini-grid’ (with spacings between
points of 2.5 pixels) around the injected source. A source is considered to be
recovered if more than 5 out of the 9 fits initialized from the mini-grid
around the injected source meet the following conditions: 1) the retrieved
centroid is less than 1 pixel from the injected location and 2) the retrieved
peak flux is within 50% of the injected value. Thus, we can place a detection
sensitivity limit at every grid point around the microlensing source star.
Note that this detection criterion is considerably more stringent than that
used for the blind search. Hence, it results in a conservative lower limit on
the brightness of recoverable sources at each location.

At the expected location of the lens, we find the $3\sigma$ flux limit to be
$H>20.57$. Along the annulus defined by $\mu_{\rm rel,hel}$, the limits are
generally better except where the annulus intersects the PSF of stars 3 and 4.
Although the limits nominally reach $H\sim 22.1$ in some places, these limits
are highly dependent on the exact level of the background in a particular
pixel, and thus, may be over-optimistic. However, stars 10 and 11 (with
magnitudes $H=21.6$ and $H=21.4$, respectively) are clearly recovered in the
blind search. Hence, we can conservatively say that the limits on the lens
flux range from 21.6 to 20.6 depending on the location along the annulus.

##  5 Discussion

Figure 4: Constraints and limits on the mass and distance of
OGLE-2007-BLG-224L. The constraints (solid lines) from $\theta_{\rm E}$ and
$\pi_{\rm E}$ from Gould et al. (2009) are shown in black and blue,
respectively, with their $1\sigma$ limits (dashed lines). The green line shows
the flux limits from MagAO after transforming to mass and distance using
stellar and sub-stellar isochrones. The dashed green line shows the limit if
an extinction of $A_{H}=0.25$ is included. Lenses to the upper left of the
green line are excluded. This limit clearly shows that only substellar lenses
(dot-dashed red line) are consistent with the microlensing measurements.

We have found no evidence for a luminous lens in OGLE-2007-BLG-224, and we
have placed upper limits on light from the lens of $H>20.57$ mag. These limits
are consistent with the prediction from Gould et al. (2009) that the lens
should be $H=25.7$. We transform this flux constraint into mass and distance
using 5-Gyr, solar-metallicity isochrones from MIST (Choi et al., 2016) for
$>0.1M_{\odot}$, BHAC15 (Baraffe et al., 2015) for $0.075$ to $0.1M_{\odot}$,
and Baraffe et al. (2003) for $<0.075M_{\odot}$ (note that because the
isochrones are not constrained to be consistent with each other, this results
in a slight hitch in the mass-luminosity relation.). In this field, Gonzalez
et al. (2011, 2012) measure $E(J-K)=0.2561$ and $A_{K}=0.1353$, so we adopt
$A_{H}=0.25$ as a reasonable estimate for the $H$-band extinction toward this
field. However, the value of the extinction and whether or not the lens
experiences some or all of it has very little effect on our conclusions.
Figure 4 shows that our limit strongly constrains the lens mass and distance,
independent of any other information. It also compares this limit to the
constraints on the lens mass and distance from $\theta_{\rm E}$ and $\pi_{\rm
E}$ assuming source distance of 7.8 kpc (Gould et al., 2009). The lack of a
detection of light from the lens in OGLE-2007-BLG-224 strongly supports the
measurement of the lens mass and distance based on the finite source and
terrestrial parallax effects measured from the light curve. This is the first
test of a terrestrial parallax measurement.

Recently, there have been several discoveries of strong free-floating planet
candidates (Mróz et al., 2017, 2018; Kim et al., 2020) including a Mars- to
Earth-mass object (Mróz et al., 2020). Proving that these planets are truly
free-floating requires demonstrating that they do not have host stars. Stellar
hosts at several to $\sim 10\ $ au for these candidates have been ruled out by
the microlensing light curves but could exist at wider separations. The limits
on lens light for OGLE-2007-BLG-224 demonstrate that current instruments can
place flux limits strong enough to rule out stellar companions to these
candidates.

We thank A. Gould for comments on the manuscript and M. van den Berg for
identifying sky fields for the MagAO observing run. KM’s work was supported by
the NASA Exoplanets Research Program (XRP) by cooperative agreement
NNX16AD44G. This work makes use of observations made with the NASA/ESA Hubble
Space Telescope, and obtained from the Hubble Legacy Archive, which is a
collaboration between the Space Telescope Science Institute (STScI/NASA), the
Space Telescope European Coordinating Facility (ST-ECF/ESA) and the Canadian
Astronomy Data Centre (CADC/NRC/CSA). It also makes use of data products from
observations made with ESO Telescopes at the La Silla Paranal Observatory
under programme ID 179.B-2002. Computations in this paper were run on the
FASRC Cannon cluster supported by the FAS Division of Science Research
Computing Group at Harvard University.

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