††thanks: A footnote to the article title

Also at ]Physics Department, XYZ University.

# Manuscript Title:
with Forced Linebreak

Nicholas Sohre  [  Stephen J. Guy<EMAIL_ADDRESS>University of Minnesota
Department of Computer Science & Engineering

###### Abstract

An article usually includes an abstract, a concise summary of the work covered
at length in the main body of the article.

Usage

Secondary publications and information retrieval purposes.

Structure

You may use the description environment to structure your abstract; use the
optional argument of the `\item` command to give the category of each item.

††preprint: APS/123-QED

##  I First-level heading:
The line break was forced via \\\

This sample document demonstrates proper use of REVTeX 4.2 (and LaTeX 2ε) in
mansucripts prepared for submission to APS journals. Further information can
be found in the REVTeX 4.2 documentation included in the distribution or
available at http://journals.aps.org/revtex/.

When commands are referred to in this example file, they are always shown with
their required arguments, using normal TeX format. In this format, `#1`, `#2`,
etc. stand for required author-supplied arguments to commands. For example, in
`\section{#1}` the `#1` stands for the title text of the author’s section
heading, and in `\title{#1}` the `#1` stands for the title text of the paper.

Line breaks in section headings at all levels can be introduced using \\\\. A
blank input line tells TeX that the paragraph has ended. Note that top-level
section headings are automatically uppercased. If a specific letter or word
should appear in lowercase instead, you must escape it using `\lowercase{#1}`
as in the word “via” above.

###  I.1 Second-level heading: Formatting

This file may be formatted in either the preprint or reprint style. reprint
format mimics final journal output. Either format may be used for submission
purposes. letter sized paper should be used when submitting to APS journals.

####  I.1.1 Wide text (A level-3 head)

The widetext environment will make the text the width of the full page, as on
page 10. (Note the use the `\pageref{#1}` command to refer to the page
number.)

##### Note (Fourth-level head is run in)

The width-changing commands only take effect in two-column formatting. There
is no effect if text is in a single column.

###  I.2 Citations and References

A citation in text uses the command `\cite{#1}` or `\onlinecite{#1}` and
refers to an entry in the bibliography. An entry in the bibliography is a
reference to another document.

####  I.2.1 Citations

Because REVTeX uses the `natbib` package of Patrick Daly, the entire
repertoire of commands in that package are available for your document; see
the `natbib` documentation for further details. Please note that REVTeX
requires version 8.31a or later of `natbib`.

##### Syntax

The argument of `\cite` may be a single _key_ , or may consist of a comma-
separated list of keys. The citation _key_ may contain letters, numbers, the
dash (-) character, or the period (.) character. New with natbib 8.3 is an
extension to the syntax that allows for a star (*) form and two optional
arguments on the citation key itself. The syntax of the `\cite` command is
thus (informally stated)

> where _optarg+key_ signifies

> where _pre_ and _post_ is whatever text you wish to place at the beginning
> and end, respectively, of the bibliographic reference (see Ref. [witten2001]
> and the two under Ref. [feyn54]). (Keep in mind that no automatic space or
> punctuation is applied.) It is highly recommended that you put the entire
> _pre_ or _post_ portion within its own set of braces, for example: `\cite`
> `{` [ `{`_text_`}`]_key_`}`. The extra set of braces will keep LaTeX out of
> trouble if your _text_ contains the comma (,) character.

The star (*) modifier to the _key_ signifies that the reference is to be
merged with the previous reference into a single bibliographic entry, a common
idiom in APS and AIP articles (see below, Ref. [epr]). When references are
merged in this way, they are separated by a semicolon instead of the period
(full stop) that would otherwise appear.

##### Eliding repeated information

When a reference is merged, some of its fields may be elided: for example,
when the author matches that of the previous reference, it is omitted. If both
author and journal match, both are omitted. If the journal matches, but the
author does not, the journal is replaced by _ibid._ , as exemplified by Ref.
[epr]. These rules embody common editorial practice in APS and AIP journals
and will only be in effect if the markup features of the APS and AIP BibTeX
styles is employed.

##### The options of the cite command itself

Please note that optional arguments to the _key_ change the reference in the
bibliography, not the citation in the body of the document. For the latter,
use the optional arguments of the `\cite` command itself: `\cite` *[_pre-
cite_][_post-cite_]`{`_key-list_`}`.

####  I.2.2 Example citations

By default, citations are numerical[Beutler1994]. Author-year citations are
used when the journal is RMP. To give a textual citation, use
`\onlinecite{#1}`: Refs. [][, andreferencestherein]witten2001, Bire82. By
default, the natbib package automatically sorts your citations into numerical
order and “compresses” runs of three or more consecutive numerical citations.
REVTeX provides the ability to automatically change the punctuation when
switching between journal styles that provide citations in square brackets and
those that use a superscript style instead. This is done through the
citeautoscript option. For instance, the journal style prb automatically
invokes this option because Physical Review B uses superscript-style
citations. The effect is to move the punctuation, which normally comes after a
citation in square brackets, to its proper position before the superscript. To
illustrate, we cite several together [[Seetheexplanationoftimetravelin]feyn54,
*[Theclassicalrelativistictreatmentof][isarelativeclassic]epr, witten2001,
Berman1983, Davies1998, Bire82], and once again in different order (Refs.
[epr, feyn54, Bire82, Berman1983, witten2001, Davies1998]). Note that the
citations were both compressed and sorted. Futhermore, running this sample
file under the prb option will move the punctuation to the correct place.

When the `prb` class option is used, the `\cite{#1}` command displays the
reference’s number as a superscript rather than in square brackets. Note that
the location of the `\cite{#1}` command should be adjusted for the reference
style: the superscript references in `prb` style must appear after
punctuation; otherwise the reference must appear before any punctuation. This
sample was written for the regular (non-prb) citation style. The command
`\onlinecite{#1}` in the prb style also displays the reference on the
baseline.

####  I.2.3 References

A reference in the bibliography is specified by a `\bibitem{#1}` command with
the same argument as the `\cite{#1}` command. `\bibitem{#1}` commands may be
crafted by hand or, preferably, generated by BibTeX. REVTeX 4.2 includes
BibTeX style files `apsrev4-2.bst`, `apsrmp4-2.bst` appropriate for Physical
Review and Reviews of Modern Physics, respectively.

####  I.2.4 Example references

This sample file employs the `\bibliography` command, which formats the
apssamp.bbl file and specifies which bibliographic databases are to be used by
BibTeX (one of these should be by arXiv convention apssamp.bib). Running
BibTeX (via bibtex apssamp) after the first pass of LaTeX produces the file
apssamp.bbl which contains the automatically formatted `\bibitem` commands
(including extra markup information via `\bibinfo` and `\bibfield` commands).
If not using BibTeX, you will have to create the `thebibiliography`
environment and its `\bibitem` commands by hand.

Numerous examples of the use of the APS bibliographic entry types appear in
the bibliography of this sample document. You can refer to the apssamp.bib
file, and compare its information to the formatted bibliography itself.

###  I.3 Footnotes

Footnotes, produced using the `\footnote{#1}` command, usually integrated into
the bibliography alongside the other entries. Numerical citation styles do
this111Automatically placing footnotes into the bibliography requires using
BibTeX to compile the bibliography.; author-year citation styles place the
footnote at the bottom of the text column. Note: due to the method used to
place footnotes in the bibliography, _you must re-run Bib TeX every time you
change any of your document’s footnotes_.

##  II Math and Equations

Inline math may be typeset using the `$` delimiters. Bold math symbols may be
achieved using the `bm` package and the `\bm{#1}` command it supplies. For
instance, a bold $\alpha$ can be typeset as `$\bm{\alpha}$` giving
$\bm{\alpha}$. Fraktur and Blackboard (or open face or double struck)
characters should be typeset using the `\mathfrak{#1}` and `\mathbb{#1}`
commands respectively. Both are supplied by the amssymb package. For example,
`$\mathbb{R}$` gives $\mathbb{R}$ and `$\mathfrak{G}$` gives $\mathfrak{G}$

In LaTeX there are many different ways to display equations, and a few
preferred ways are noted below. Displayed math will center by default. Use the
class option `fleqn` to flush equations left.

Below we have numbered single-line equations; this is the most common type of
equation in Physical Review:

$\displaystyle\chi_{+}(p)\lesssim{\bf[}2|{\bf p}|(|{\bf
p}|+p_{z}){\bf]}^{-1/2}\left(\begin{array}[]{c}|{\bf p}|+p_{z}\\\
px+ip_{y}\end{array}\right)\;,$ (3) $\displaystyle\left\\{\openone
234567890abc123\alpha\beta\gamma\delta
1234556\alpha\beta\frac{1\sum^{a}_{b}}{A^{2}}\right\\}.$ (4)

Note the open one in Eq. (4).

Not all numbered equations will fit within a narrow column this way. The
equation number will move down automatically if it cannot fit on the same line
with a one-line equation:

$\left\\{ab12345678abc123456abcdef\alpha\beta\gamma\delta
1234556\alpha\beta\frac{1\sum^{a}_{b}}{A^{2}}\right\\}.$ (5)

When the `\label{#1}` command is used [cf. input for Eq. (4)], the equation
can be referred to in text without knowing the equation number that TeX will
assign to it. Just use `\ref{#1}`, where `#1` is the same name that used in
the `\label{#1}` command.

Unnumbered single-line equations can be typeset using the `\[`, `\]` format:

$g^{+}g^{+}\rightarrow
g^{+}g^{+}g^{+}g^{+}\dots~{},~{}~{}q^{+}q^{+}\rightarrow
q^{+}g^{+}g^{+}\dots~{}.$

###  II.1 Multiline equations

Multiline equations are obtained by using the `eqnarray` environment. Use the
`\nonumber` command at the end of each line to avoid assigning a number:

$\displaystyle{\cal M}=$ $\displaystyle
ig_{Z}^{2}(4E_{1}E_{2})^{1/2}(l_{i}^{2})^{-1}\delta_{\sigma_{1},-\sigma_{2}}(g_{\sigma_{2}}^{e})^{2}\chi_{-\sigma_{2}}(p_{2})$
(6)
$\displaystyle\times[\epsilon_{j}l_{i}\epsilon_{i}]_{\sigma_{1}}\chi_{\sigma_{1}}(p_{1}),$
$\displaystyle\sum|M^{\text{viol}}_{g}|^{2}$ $\displaystyle=$ $\displaystyle
g^{2n-4}_{S}(Q^{2})~{}N^{n-2}(N^{2}-1)$ (7)
$\displaystyle\times\left(\sum_{i<j}\right)\sum_{\text{perm}}\frac{1}{S_{12}}\frac{1}{S_{12}}\sum_{\tau}c^{f}_{\tau}~{}.$

Note: Do not use `\label{#1}` on a line of a multiline equation if `\nonumber`
is also used on that line. Incorrect cross-referencing will result. Notice the
use `\text{#1}` for using a Roman font within a math environment.

To set a multiline equation without _any_ equation numbers, use the
`\begin{eqnarray*}`, `\end{eqnarray*}` format:

$\displaystyle\sum|M^{\text{viol}}_{g}|^{2}$ $\displaystyle=$ $\displaystyle
g^{2n-4}_{S}(Q^{2})~{}N^{n-2}(N^{2}-1)$
$\displaystyle\times\left(\sum_{i<j}\right)\left(\sum_{\text{perm}}\frac{1}{S_{12}S_{23}S_{n1}}\right)\frac{1}{S_{12}}~{}.$

To obtain numbers not normally produced by the automatic numbering, use the
`\tag{#1}` command, where `#1` is the desired equation number. For example, to
get an equation number of (8),

$g^{+}g^{+}\rightarrow
g^{+}g^{+}g^{+}g^{+}\dots~{},~{}~{}q^{+}q^{+}\rightarrow
q^{+}g^{+}g^{+}\dots~{}.$ (8)

##### A few notes on tags

`\tag{#1}` requires the amsmath package. Place the `\tag{#1}` command before
the `\label{#1}`, if any. The numbering produced by `\tag{#1}` does not affect
the automatic numbering in REVTeX; therefore, the number must be known ahead
of time, and it must be manually adjusted if other equations are added.
`\tag{#1}` works with both single-line and multiline equations. `\tag{#1}`
should only be used in exceptional cases—do not use it to number many
equations in your paper. Please note that this feature of the amsmath package
is _not_ compatible with the hyperref (6.77u) package.

Enclosing display math within `\begin{subequations}` and `\end{subequations}`
will produce a set of equations that are labeled with letters, as shown in
Eqs. (9b) and (9a) below. You may include any number of single-line and
multiline equations, although it is probably not a good idea to follow one
display math directly after another.

$\displaystyle{\cal M}=$ $\displaystyle
ig_{Z}^{2}(4E_{1}E_{2})^{1/2}(l_{i}^{2})^{-1}(g_{\sigma_{2}}^{e})^{2}\chi_{-\sigma_{2}}(p_{2})$
(9a) $\displaystyle\times[\epsilon_{i}]_{\sigma_{1}}\chi_{\sigma_{1}}(p_{1}).$
$\left\\{abc123456abcdef\alpha\beta\gamma\delta
1234556\alpha\beta\frac{1\sum^{a}_{b}}{A^{2}}\right\\},$ (9b)

Giving a `\label{#1}` command directly after the `\begin{subequations}`,
allows you to reference all the equations in the subequations environment. For
example, the equations in the preceding subequations environment were Eqs.
(9).

####  II.1.1 Wide equations

The equation that follows is set in a wide format, i.e., it spans the full
page. The wide format is reserved for long equations that cannot easily be set
in a single column:

${\cal
R}^{(\text{d})}=g_{\sigma_{2}}^{e}\left(\frac{[\Gamma^{Z}(3,21)]_{\sigma_{1}}}{Q_{12}^{2}-M_{W}^{2}}+\frac{[\Gamma^{Z}(13,2)]_{\sigma_{1}}}{Q_{13}^{2}-M_{W}^{2}}\right)+x_{W}Q_{e}\left(\frac{[\Gamma^{\gamma}(3,21)]_{\sigma_{1}}}{Q_{12}^{2}-M_{W}^{2}}+\frac{[\Gamma^{\gamma}(13,2)]_{\sigma_{1}}}{Q_{13}^{2}-M_{W}^{2}}\right)\;.$
(10)

This is typed to show how the output appears in wide format. (Incidentally,
since there is no blank line between the equation environment above and the
start of this paragraph, this paragraph is not indented.)

##  III Cross-referencing

REVTeX will automatically number such things as sections, footnotes,
equations, figure captions, and table captions. In order to reference them in
text, use the `\label{#1}` and `\ref{#1}` commands. To reference a particular
page, use the `\pageref{#1}` command.

The `\label{#1}` should appear within the section heading, within the footnote
text, within the equation, or within the table or figure caption. The
`\ref{#1}` command is used in text at the point where the reference is to be
displayed. Some examples: Section I on page I, Table 1,

Table 1: A table that fits into a single column of a two-column layout. Note that REVTeX 4 adjusts the intercolumn spacing so that the table fills the entire width of the column. Table captions are numbered automatically. This table illustrates left-, center-, decimal- and right-aligned columns, along with the use of the ruledtabular environment which sets the Scotch (double) rules above and below the alignment, per APS style.  Left222Note a. | Centered333Note b. | Decimal | Right
---|---|---|---
1 | 2 | $3.001$ | 4
10 | 20 | $30$ | 40
100 | 200 | $300.0$ | 400

and Fig. 1.

Figure 1:  A figure caption. The figure captions are automatically numbered.

##  IV Floats: Figures, Tables, Videos, etc.

Figures and tables are usually allowed to “float”, which means that their
placement is determined by LaTeX, while the document is being typeset.

Use the figure environment for a figure, the table environment for a table. In
each case, use the `\caption` command within to give the text of the figure or
table caption along with the `\label` command to provide a key for referring
to this figure or table. The typical content of a figure is an image of some
kind; that of a table is an alignment.

Figure 2: Use the figure* environment to get a wide figure that spans the page in twocolumn formatting. Table 2: This is a wide table that spans the full page width in a two-column layout. It is formatted using the table* environment. It also demonstates the use of \multicolumn in rows with entries that span more than one column. | $D_{4h}^{1}$ | $D_{4h}^{5}$
---|---|---
Ion | 1st alternative | 2nd alternative | lst alternative | 2nd alternative
K | $(2e)+(2f)$ | $(4i)$ | $(2c)+(2d)$ | $(4f)$
Mn |  $(2g)$444The $z$ parameter of these positions is $z\sim\frac{1}{4}$. | $(a)+(b)+(c)+(d)$ | $(4e)$ | $(2a)+(2b)$
Cl | $(a)+(b)+(c)+(d)$ |  $(2g)$111Here’s the first, from Ref. feyn54. | $(4e)^{\text{a}}$ |
He | $(8r)^{\text{a}}$ | $(4j)^{\text{a}}$ | $(4g)^{\text{a}}$ |
Ag |  | $(4k)^{\text{a}}$ |  | $(4h)^{\text{a}}$

Insert an image using either the graphics or graphix packages, which define
the `\includegraphics{#1}` command. (The two packages differ in respect of the
optional arguments used to specify the orientation, scaling, and translation
of the image.) To create an alignment, use the tabular environment.

The best place to locate the figure or table environment is immediately
following its first reference in text; this sample document illustrates this
practice for Fig. 1, which shows a figure that is small enough to fit in a
single column.

In exceptional cases, you will need to move the float earlier in the document,
as was done with Table 2: LaTeX’s float placement algorithms need to know
about a full-page-width float earlier.

Fig. 2 has content that is too wide for a single column, so the figure*
environment has been used.

Table 3: Numbers in columns Three–Five are aligned with the “d” column specifier (requires the dcolumn package). Non-numeric entries (those entries without a “.”) in a “d” column are aligned on the decimal point. Use the “D” specifier for more complex layouts.  One | Two | Three | Four | Five
---|---|---|---|---
one | two | three | four | five
He | 2 | $2.77234$ | $45672.$ | $0.69$
C555Some tables require footnotes. | C666Some tables need more than one footnote. | $12537.64$ | $37.66345$ | $86.37$

The content of a table is typically a tabular environment, giving rows of type
in aligned columns. Column entries separated by `&`’s, and each row ends with
\\\\. The required argument for the tabular environment specifies how data are
aligned in the columns. For instance, entries may be centered, left-justified,
right-justified, aligned on a decimal point. Extra column-spacing may be be
specified as well, although REVTeX 4 sets this spacing so that the columns
fill the width of the table. Horizontal rules are typeset using the `\hline`
command. The doubled (or Scotch) rules that appear at the top and bottom of a
table can be achieved enclosing the tabular environment within a ruledtabular
environment. Rows whose columns span multiple columns can be typeset using the
`\multicolumn{#1}{#2}{#3}` command (for example, see the first row of Table
2).

Tables 1, 2, 3, and 4

Table 4:  A table with numerous columns that still fits into a single column. Here, several entries share the same footnote. Inspect the LaTeX input for this table to see exactly how it is done. |  $r_{c}$ (Å) |  $r_{0}$ (Å) | $\kappa r_{0}$ |  |  $r_{c}$ (Å) |  $r_{0}$ (Å) | $\kappa r_{0}$
---|---|---|---|---|---|---|---
Cu | 0.800 | 14.10 | 2.550 | Sn111Here’s the first, from Ref. feyn54. | 0.680 | 1.870 | 3.700
Ag | 0.990 | 15.90 | 2.710 | Pb222Here’s the second. | 0.450 | 1.930 | 3.760
Au | 1.150 | 15.90 | 2.710 | Ca333Here’s the third. | 0.750 | 2.170 | 3.560
Mg | 0.490 | 17.60 | 3.200 | Sr444Here’s the fourth. | 0.900 | 2.370 | 3.720
Zn | 0.300 | 15.20 | 2.970 | Li222Here’s the second. | 0.380 | 1.730 | 2.830
Cd | 0.530 | 17.10 | 3.160 | Na555And etc. | 0.760 | 2.110 | 3.120
Hg | 0.550 | 17.80 | 3.220 | K555And etc. | 1.120 | 2.620 | 3.480
Al | 0.230 | 15.80 | 3.240 | Rb333Here’s the third. | 1.330 | 2.800 | 3.590
Ga | 0.310 | 16.70 | 3.330 | Cs444Here’s the fourth. | 1.420 | 3.030 | 3.740
In | 0.460 | 18.40 | 3.500 | Ba555And etc. | 0.960 | 2.460 | 3.780
Tl | 0.480 | 18.90 | 3.550 |  |  |  |

show various effects. A table that fits in a single column employs the table
environment. Table 2 is a wide table, set with the table* environment. Long
tables may need to break across pages. The most straightforward way to
accomplish this is to specify the `[H]` float placement on the table or table*
environment. However, the LaTeX 2ε package longtable allows headers and
footers to be specified for each page of the table. A simple example of the
use of longtable can be found in the file summary.tex that is included with
the REVTeX 4 distribution.

There are two methods for setting footnotes within a table (these footnotes
will be displayed directly below the table rather than at the bottom of the
page or in the bibliography). The easiest and preferred method is just to use
the `\footnote{#1}` command. This will automatically enumerate the footnotes
with lowercase roman letters. However, it is sometimes necessary to have
multiple entries in the table share the same footnote. In this case, there is
no choice but to manually create the footnotes using `\footnotemark[#1]` and
`\footnotetext[#1]{#2}`. #1 is a numeric value. Each time the same value for
#1 is used, the same mark is produced in the table. The
`\footnotetext[#1]{#2}` commands are placed after the tabular environment.
Examine the LaTeX source and output for Tables 1 and 4 for examples.

Video IV illustrates several features new with REVTeX4.2, starting with the
video environment, which is in the same category with figure and table.
http://link.aps.org/multimedia/PRSTPER/v4/i1/e010101Students explain their
initial idea about Newton’s third law to a teaching assistant. Clip (a): same
force. Clip (b): move backwards.  The `\setfloatlink` command causes the title
of the video to be a hyperlink to the indicated URL; it may be used with any
environment that takes the `\caption` command. The `\href` command has the
same significance as it does in the context of the hyperref package: the
second argument is a piece of text to be typeset in your document; the first
is its hyperlink, a URL.

Physical Review style requires that the initial citation of figures or tables
be in numerical order in text, so don’t cite Fig. 2 until Fig. 1 has been
cited.

###### Acknowledgements.

We wish to acknowledge the support of the author community in using REVTeX,
offering suggestions and encouragement, testing new versions, ….

##  Appendix A Appendixes

To start the appendixes, use the `\appendix` command. This signals that all
following section commands refer to appendixes instead of regular sections.
Therefore, the `\appendix` command should be used only once—to setup the
section commands to act as appendixes. Thereafter normal section commands are
used. The heading for a section can be left empty. For example,

    \appendix
    \section{}

will produce an appendix heading that says “APPENDIX A” and

    \appendix
    \section{Background}

will produce an appendix heading that says “APPENDIX A: BACKGROUND” (note that
the colon is set automatically).

If there is only one appendix, then the letter “A” should not appear. This is
suppressed by using the star version of the appendix command (`\appendix*` in
the place of `\appendix`).

##  Appendix B A little more on appendixes

Observe that this appendix was started by using

    \section{A little more on appendixes}

Note the equation number in an appendix:

$E=mc^{2}.$ (11)

###  B.1 A subsection in an appendix

You can use a subsection or subsubsection in an appendix. Note the numbering:
we are now in Appendix B.1.

Note the equation numbers in this appendix, produced with the subequations
environment:

$\displaystyle E$ $\displaystyle=$ $\displaystyle mc,$ (12a) $\displaystyle E$
$\displaystyle=$ $\displaystyle mc^{2},$ (12b) $\displaystyle E$
$\displaystyle\gtrsim$ $\displaystyle mc^{3}.$ (12c)

They turn out to be Eqs. (12a), (12b), and (12c).


# Asymptotic bounds for cycle integrals of the $j$-function on Markov
geodesics

Paloma Bengoechea  ETH Zurich
Department of Mathematics
Ramistrasse 101
8092 Zurich
Switzerland<EMAIL_ADDRESS>
###### Abstract.

We give asymptotic upper and lower bounds for the real and imaginary parts of
cycle integrals of the classical modular $j$-function along geodesics that
correspond to Markov irrationalities.

Bengoechea’s research is supported by SNF grant 173976.

##  1\. Introduction

Let $w$ be a real quadratic irrationality, so a root of an equation

$ax^{2}+bx+c=0\qquad(a,b,c\in\mathbb{Z},\quad(a,b,c)=1)$

with positive non square discriminant $D=b^{2}-4ac$. Let $Q$ be the quadratic
form $[a,b,c]$. The geodesic $S_{Q}$ in the upper half plane $\mathcal{H}$
joining $w$ and its Galois conjugate is given by the equation

$a|z|^{2}+b\,\mathrm{Re}\,z+c=0\qquad(z\in\mathcal{H}).$

We orientate $S_{Q}$ counterclockwise if $a>0$ and clockwise if $a<0$. There
is an infinite cyclic group $\Gamma_{Q}$ in $\mathrm{SL}(2,\mathbb{Z})$,
corresponding to the group of totally positive units in
$\mathbb{Q}(\sqrt{D})$, that preserves $Q$ and hence $S_{Q}$. The smallest
positive unit in $\mathbb{Q}(\sqrt{D})$ is given by
$\varepsilon=(t+u\sqrt{D})/2$, where $(t,u)$ is the smallest positive integral
solution to Pell’s equation $t^{2}-Du^{2}=4$. The induced geodesic
$C_{Q}=\Gamma_{Q}\backslash S_{Q}$ on the modular surface is closed,
primitive, positively-oriented with length

$\mathrm{length}(C_{Q})=\int_{C_{Q}}\frac{\sqrt{D}}{Q(z,1)}dz=2\log\varepsilon.$

Here $\frac{\sqrt{D}}{Q(z,1)}dz$ is the hyperbolic arc length on $S_{Q}$. As
usual, we denote by $j(z)$ the modular function for
$\mathrm{SL}(2,\mathbb{Z})$ introduced by Klein as an invariant of elliptic
curves. The group $\Gamma_{Q}$ preserves $j(z)\frac{\sqrt{D}}{Q(z,1)}dz$, and
so we can consider the integral

$\int_{C_{Q}}j(z)\frac{\sqrt{D}}{Q(z,1)}dz$

and define the ‘value’ (also called cycle integral) of $j$ at $w$, as the
complex number:

(1.1)
$j(w)=j(Q):=\dfrac{1}{2\log\varepsilon}\int_{C_{Q}}j(z)\frac{\sqrt{D}}{Q(z,1)}dz.$

Note that the integral above is $\mathrm{SL}(2,\mathbb{Z})$-invariant.

Cycle integrals have been related to mock modular forms [11], to modular knots
[13] and to class numbers of real quadratic fields [14]. Moreover, cycle
integrals of the Klein invariant $j$ share several analogies with singular
moduli (the values of the $j$-function at imaginary quadratic irrationalities)
when both are gathered in ‘traces’ (see [11], [12], [10], [20]). By analogy to
the traces of singular moduli (the values of the $j$-function at imaginary
quadratic irrationalities) we define $\mbox{Tr}_{D}j:=\sum j(w_{Q})$ where the
sum is over $\mathrm{SL}(2,\mathbb{Z})$-classes of indefinite binary quadratic
forms $Q$ of fundamental discriminant $D>0$. The asymptotic distribution of
the traces was studied in [9] and [10],[20] for negative and positive
discriminants respectively. As fundamental discriminants
$D\rightarrow+\infty$, it was shown in [10],[20] that

(1.2) $\frac{\mbox{Tr}_{D}(j)}{\mbox{Tr}_{D}(1)}\rightarrow 720.$

Here $720$ is an ‘average’ value of the $j$ function. The individual values
remain very much unknown. In [3] we gave the first bounds for the real parts
of the values $j(w)$ (in fact $j$ could be replaced by any modular function
$f$ which is real valued on the geodesic arc
$\left\\{e^{i\theta}:\pi/3\leq\theta\leq 2\pi/3\right\\}$). We showed that
$\mathrm{Re}(j(w))\leq 744$ (we recall that $744$ is the constant term in the
Fourier expansion of $j$). For quadratic irrationalities $w$ satisfying a
‘quite strong’ diophantine condition, we proved that $\mathrm{Re}(j(w))\geq
j(\frac{1+\sqrt{5}}{2})\approx 706.3248$. These bounds are optimal and were
conjectured earlier by Kaneko in [16] based on numerical evidence. The
imaginary parts of $j(w)$ are conjectured to lie in $(-1,1)$, but to our
knowledge nothing is known yet.

The values $j(w)$ are particularly interesting at Markov irrationalities $w$.
Markov irrationalities are important in diophantine approximation and in the
search of positive minima of indefinite binary quadratic forms. They are
structured on a tree called the Markov-Hurwitz tree and there is a very rich
theory attached to them in the interplay of number theory, diophantine
approximation, hyperbolic geometry, dynamics and graph theory. Kaneko
published in [16] the first numerical data on values $j(w)$ at Markov
irrationalities, together with some conjectures based on his data. Namely, he
conjectured that each $j(w)$ is between the values of $j$ at two Markov
irrationalities that lie above $w$ on the tree. We call these two
irrationalities the predecessors of $w$ and we call call this property
‘interlacing property’. In [2] we proved that for every branch of the tree,
the interlacing property holds after some level that depends on the branch.
One can think of that result as a ‘local’ asymptotic interlacing property (an
asymptotic property that holds for each branch).

In this paper we prove the ‘global’ interlacing property, namely:

######  Theorem 1.

Let $w=w_{n}$ be a Markov irrationality and $n$ be the level of $w$ in the
Markov-Hurwitz tree. There is a unique path on the Markov-Hurwitz tree that
ends with $w_{n}$; let $w_{n-1}$ be the Markov irrationality above $w_{n}$ on
the path, and $w_{n^{\prime}-1}$ be the Markov irrationality above the last
irrationality where the path turns. There exists an integer $n_{0}$ such that,
if $n>n_{0}$, then $j(w_{n})$ lies between $j(w_{n-1})$ and
$j(w_{n^{\prime}-1})$.

The irrationalities $w_{n-1}$ and $w_{n^{\prime}-1}$ are the ‘predecessors’ of
$w_{n}$ in Theorem 1. The interlacing property is expected to hold for $n\geq
2$, hence we expect the bounds:

$\displaystyle 706.32481\approx j((1+\sqrt{5})/2)\leq$
$\displaystyle\mathrm{Re}(j(w))\leq j(\sqrt{2})\approx 709.8929$
$\displaystyle-0.26703\ldots\leq$ $\displaystyle\mathrm{Im}(j(w))\leq
0.26703\ldots.$

By making explicit most of the estimations in Theorem 1, we obtain:

######  Theorem 2.

Let $w$ be a Markov irrationality and $n$ be the level of $w$ in the Markov-
Hurwitz tree. We have that

$\displaystyle 681.50081\leq\mathrm{Re}(j(w))$ $\displaystyle\leq
742.03641\qquad as\ n\rightarrow\infty$
$\displaystyle-0.93637\leq\mathrm{Im}(j(w))$ $\displaystyle\leq 0.67396\qquad
as\ n\rightarrow\infty.$

Although the bounds in Theorem 2 are not optimal, they are the first lower
bounds for the real parts of the values $j(w)$ at Markov irrationalities and
the first bounds for the imaginary parts that we have until now.

One reason why we obtained only a ‘local’ interlacing property in [2] is that
we lacked of a convenient order on the tree. A new key approach here is to
order the Markov-Hurwitz tree by Farey fractions (which can probably be more
exploited in the future). We also exploit more accurately the diophantine
properties of the Markov-Hurwitz tree as well as its relations with the Farey
tree, and use an asymptotic formula from [23] on Markov numbers. The arguments
used in this paper apply to any modular function $f$ which is real valued on
the geodesic arc $\left\\{e^{i\theta}:\pi/3\leq\theta\leq 2\pi/3\right\\}$.
The values (7.4), (10.2), (10.3) will change when we replace $j$ by $f$, and
so will the subsequent calculations.

The paper is organized as follows. In the next two sections, we quickly review
the main key points of Markov’s theory and other related facts that will be
useful. Concretely, in section 2, we introduce Markov numbers and Markov
irrationalities. In section 3 we introduce the Markov-Hurwitz and the Farey
trees as well as some of their properties and interrelations. In section 4 we
give an asymptotic formula for $\log\varepsilon$ in terms of the Farey tree by
using Zagier’s asymptotic formula for Markov numbers in [23]. In section 5 we
write the values $j(w)$ in terms of cycles of certain reduced binary quadratic
forms, usually known as ‘simple forms’ after Zagier. Simple forms can be
written themselves in terms of cycles of quadratic irrationalities produced by
a certain continued fraction algorithm. We work with the cycles of continued
fraction expansions in sections 6 and 7 to obtain a ‘local’ formula for
$(2\log\varepsilon)j(w)$ that depends on two neighbours of $w$ on the Markov-
Hurwitz tree (the two predecessors of $w$). In section 8 we deduce a ‘global’
formula for $(2\log\varepsilon)j(w)$ from the local formula obtained in
section 7. We prove Theorem 1 in section 9 using the global formula from
section 8 and the asymptotic formula for $\log\varepsilon$ given in section 4.
We prove Theorem 2 in section 10. We give some numerical data in the appendix,
in section 11.

##  2\. Markov’s theory

Markov’s work (1880, [18], [19]) establishes very beautiful connections
between positive integral minima for indefinite binary quadratic forms and the
Lagrange-Hurwitz problem in diophantine approximation. The Lagrange-Hurwitz
problem consists in describing the Lagrange constants

$L(x)=(\liminf_{q\rightarrow\infty}q\|qx\|),$

where $x$ runs through the real numbers and $\|\cdot\|$ denotes the distance
to a closest integer. The quantity $L(x)$ provides a ‘measure’ of how well $x$
can be approximated by the rationals. For almost all $x\in\mathbb{R}$ we have
$L(x)=0$, and when $L(x)>0$ we call $x$ _badly approximable_. A well known
theorem of Hurwitz states that $L(x)<1/\sqrt{5}$.

The Lagrange spectrum
$L:=\left\\{L(x)^{-1}\right\\}_{x\in\mathbb{R}}\subseteq[\sqrt{5},\infty]$ is
structured in three parts: $\L\cap[\sqrt{5},3)$ is discrete with $3$ as the
only accumulation point, $\L\cap(3,F]$ is fractal, where $F\approx 4.528$ is
Freiman’s constant, and $\L\cap(F,\infty]$ is continuous.

Markov discovered that on the interval $(0,3)$ the Lagrange spectrum and the
spectrum $M$ of integral minima

$M(Q)=\dfrac{\sqrt{D}}{\underset{\begin{subarray}{c}(x,y)\in\mathbb{Z}^{2}\\\
(x,y)\neq(0,0)\end{subarray}}{\mathrm{\inf}}|Q(x,y)|}$

of binary quadratic forms

$Q(x,y)=Ax^{2}+Bxy+Cy^{2}\qquad(A,B,C\in\mathbb{R})$

with positive discriminant $D=B^{2}-4AC$, are the same.

The unifying thread is the diophantine equation

(2.1) $a^{2}+b^{2}+c^{2}=3abc\qquad(a,b,c\in\mathbb{N}),$

and its integer solutions. The integer solutions $(a,b,c)$ are obtained by
starting with $(1,1,1)$, $(1,1,2)$, $(2,1,5)$ and then proceeding recursively
going from $(a,b,c)$ to the new triples obtained by Vieta involutions
$(c,b,3bc-a)$ and $(a,c,3ac-b)$. _Markov numbers_ are the greatest coordinates
of each solution $(a,b,c)$. They form the _Markov sequence_

(2.2)
$\left\\{c_{i}\right\\}_{i=1}^{\infty}=\left\\{1,2,5,13,29,34,89,169,194,\ldots\right\\}.$

(We count multiplicities if the unicity conjecture is false.) To each Markov
number $c$, a _Markov irrationality_

$w=\dfrac{3c-2k+\sqrt{9c^{2}-4}}{2c}$

is associated, where $k$ is an integer that satisfies $ak\equiv b\pmod{c}$,
$0\leq k<c$, and $(a,b,c)$ is a solution to (2.1) with $c$ maximal. The
quadratic $w$ is a root of the Markov form

$Q(x,y)=cx^{2}+(3c-2k)xy+(\ell-3k)y^{2},$

where $\ell=\frac{k^{2}+1}{c}\in\mathbb{Z}$. Then

$M(Q)=L(w)^{-1}=\sqrt{9-4/c^{2}}.$

Moreover,

$M\cap(0,3)=\L\cap(0,3)=\left\\{\sqrt{9-4/c_{i}^{2}}\right\\}_{i\geq 1}$

and any $x\in\mathbb{R}$ with $L(x)^{-1}\in(0,3)$ or any $Q$ with
$M(Q)\in(0,3)$ is $\mathrm{GL}(2,\mathbb{Z})$-equivalent to a Markov quadratic
or a Markov form respectively.

##  3\. Trees related to Markov’s theory

###  3.1. The Markov-Hurwitz tree

The solutions of the diophantine equation (2.1) inherit a tree structure with
two bifurcations from the two Vieta involutions described earlier:

[ $(1,1,1)$ $(1,1,2)$ [ $\begin{array}[]{c}(2,1,5)\end{array}$ [
$\begin{array}[]{c}(5,1,13)\end{array}$ [
$\begin{array}[]{c}(13,1,34)\end{array}$ [] []] [
$\begin{array}[]{c}(5,13,194)\end{array}$ [] [] ]] [
$\begin{array}[]{c}(2,5,29)\end{array}$ [
$\begin{array}[]{c}(29,5,433)\end{array}$ [] []] [
$\begin{array}[]{c}(2,29,169)\end{array}$ [] []]]]]

Figure 1. The Markov tree

Naturally, we can consider the parallel tree of Markov irrationalities, where
each vertex is a quadratic irrationality that corresponds to a Markov number.
What is more interesting is that each Markov irrationality can be constructed
from two predecessors on the tree by a simple conjunction operation on the
continued fraction expansions. A formal decription of the procedure using ‘+’
continued fraction expansions can be found in [4]. It will be more convenient
for us to work with the ‘–’ continued fraction since it corresponds to
transformations by the modular group (whereas the ‘+’ continued fraction
corresponds to transformations by $\mathrm{GL}(2,\mathbb{Z})$). It was
observed in [2] that roughly the same conjunction operation works with the ‘–’
continued fraction. We denote by

$(a_{1},a_{2},a_{3},\ldots)=a_{1}-\dfrac{1}{a_{2}-\dfrac{1}{a_{3}-\dfrac{1}{\ddots}}}$

the ‘–’ continued fraction expansion, with $a_{i}\in\mathbb{Z}$ and $a_{i}\geq
2$ for $i\geq 2$.

[ $(\overline{3})$ $(\overline{2,4})$ [ $(\overline{2,3,4})$ [
$(\overline{2,3_{2},4})$ [ $(\overline{2,3_{3},4})$ [$\ldots$] [$\ldots$]] [
$(\overline{2,3,4,2,3_{2},4})$ [$\ldots$] [$\ldots$] ]] [
$(\overline{2,4,2,3,4})$ [ $(\overline{2,4,(2,3,4)_{2}})$ [$\ldots$]
[$\ldots$]] [ $(\overline{(2,4)_{2},2,3,4})$ [$\ldots$] [$\ldots$]]]]]

Figure 2. The Markov-Hurwitz tree $\mathcal{MH}$

We define the conjunction operation of two periods as

(3.1)
$(\overline{a_{1},\ldots,a_{r}})\odot(\overline{b_{1},\ldots,b_{s}})=(\overline{a_{1},\ldots,a_{r},b_{1},\ldots,b_{s}}).$

Each Markov irrationality not on the most left branch is the result of the
conjunction operation of two predecessors: its immediate predecessor on the
same branch and the immediate predecessor of the tip of the branch. We call
right predecessor the one on the right of $w$ on the tree, and left
predecessor the one on the left. With this terminology, $w$ is the result of
the cojunction operation of the right predecessor with the left predecessor.
For example,

$(\overline{2,4,(2,3,4)_{2}})=(\overline{2,4,2,3,4})\cdot(\overline{2,3,4});$

the right and left predecessors are $(\overline{2,4,2,3,4})$ and
$(\overline{2,3,4})$ respectively. On the most left branch, the Markov
irrationality at level $n$ is $(\overline{2,3_{n},4})$. We call
$(\overline{3})$ the left predecessor and $(\overline{2,3_{n-1},4})$ the right
predecessor.

###  3.2. The Farey tree

There is a natural parametrisation of Markov numbers by Farey fractions which
goes back to Frobenius [15]. The Farey tree is constructed by following a very
similar procedure as described in section 3.1 for the Markov-Hurwitz tree. We
start with $\frac{0}{1}$, $\frac{1}{2}$ and $\frac{1}{3}$, and construct each
following fraction from the two predecessors $\frac{a}{b},\frac{c}{d}$ that
are in the same position as in the Markov-Hurwitz tree ($\frac{a}{b}$ is the
right predecessor and $\frac{c}{d}$ is the left predecessor) by taking the
Farey median

$\dfrac{a}{b}*\dfrac{c}{d}=\dfrac{a+c}{b+d}.$

Hence each vertex of the tree is a Farey fraction $p/q$ that is in
correspondence with a Markov quadratic $w=w(p/q)$ and a Markov number
$c=m(p/q)$. We denote the Farey tree by $\mathcal{F}$.

[ $\dfrac{0}{1}$ $\dfrac{1}{2}$ [ $\dfrac{1}{3}$ [ $\dfrac{1}{4}$ [
$\dfrac{1}{5}$ [] []] [ $\dfrac{2}{7}$ [] [] ]] [ $\dfrac{2}{5}$ [
$\dfrac{3}{8}$ [] []] [ $\dfrac{3}{7}$ [] []]]]]

Figure 3. The Farey tree $\mathcal{F}$

By the construction of the Markov-Hurwitz and the Farey trees, the
denominators in $\mathcal{F}$ correspond to the lengths of the periods of the
continued fractions that occupy the same corresponding position in
$\mathcal{MH}$.

The operation giving the denominators in the Farey tree corresponds to the
well known Euclidean algorithm which, starting with $(0,1,1)$, gives from a
triple $(s,t,u)$ two new triples $(s,u,s+u)$ and $(t,u,t+u)$, obtaining thus
all solutions to the equation

(3.2) $s+t=u,\quad 0\leq s\leq t\leq u,\quad(s,t)=1.$

The denominators in the Farey tree are the maximum terms of the solutions
$(s,t,u)$.

##  4\. Asymptotic formula for $\log\varepsilon$ in terms of Farey
denominators

We order the denominators of the Farey fractions in $\mathcal{F}$ as real
increasing numbers $(q_{n})_{n\geq 1}$. If two Farey fractions
$\frac{p}{q},\frac{p^{\prime}}{q^{\prime}}$ satisfy $q=q^{\prime}$, then we
write $q=q_{n}$, $q^{\prime}=q_{m}$ with $n<m$ if
$m(\frac{p}{q})<m(\frac{p^{\prime}}{q^{\prime}})$. Hence we obtain a sequence:

(4.1) $q_{1}=1,\quad q_{2}=2,\quad q_{3}=3,\quad q_{4}=4,\quad q_{5}=5,\quad
q_{6}=5,\ldots$

that is in bijective correspondence with the sequence of Markov numbers (2.2).

The number of denominators $q_{n}$ less than $x$ (with multiplicity) is given
by

$\\#\left\\{q_{n}\leq x\right\\}_{n\geq 1}=\\#\left\\{(s,t,u)\ solution\ to\
\eqref{Euclideq}:\,u\leq x\right\\}=1+\dfrac{1}{2}\sum_{u\leq x}\varphi(u),$

where $\varphi$ is the Euler function. From the asymptotic formula

$1+\dfrac{1}{2}\sum_{u\leq x}\varphi(u)\sim\dfrac{3}{2\pi^{2}}x^{2},$

it follows that

(4.2) $q_{n}\sim\pi\sqrt{\frac{2}{3}n}.$

Let $w=w(p/q)$ be a Markov irrationality and $p/q$ be its corresponding Farey
fraction. Set $q=q_{n}$, where $q_{n}$ is a Farey denominator ordered as in
(4.1), so that $m(p/q)=c_{n}$. The discriminant of $w$ is $9c_{n}^{2}-4$, so
Pell’s equation is

$t^{2}-(9c_{n}^{2}-4)u^{2}=4,$

with $(3c_{n},1)$ being the smallest positive solution. Thus
$\varepsilon=\frac{3c_{n}+\sqrt{9c_{n}^{2}-4}}{2}$.

By a result of Zagier [23], we have that

(4.3) $c_{n}\sim e^{\sqrt{\frac{n}{C}}},$

where $C\approx 0.18071704711507$. Using (4.2),

(4.4) $c_{n}\sim e^{\frac{q_{n}}{\pi}\sqrt{\frac{3}{2C}}}.$

Hence

(4.5)
$\log\varepsilon\sim\dfrac{\sqrt{3}}{\sqrt{2C}\pi}q_{n}+\log\frac{3}{2}.$

##  5\. Simple forms and continued fractions

Any indefinite binary quadratic form $[a,b,c]$ is
$\mathrm{SL}(2,\mathbb{Z})$-equivalent to one satisfying

(5.1) $a>0>c.$

There is a finite number of forms satisfying (5.1) with fixed discriminant.
Such forms are commonly called ‘simple’ after Zagier [22] and play an
important role in the theory of rational periods and period functions of
modular forms (see [17], [24], [5], [1]).

It is shown in [6] that all simple forms $\mathrm{SL}(2,\mathbb{Z})$
equivalent to a form $[a,b,c]$ are obtained by applying iteratively the
following continued fraction algorithm. Let $w$ be the period in the ‘–’
continued fraction expansion of the root $\frac{-b+\sqrt{b^{2}-4ac}}{2a}$; we
define

(5.2) $w^{(1)}=w-1,\qquad
w^{(i+1)}=\left\\{\begin{array}[]{ll}w^{(i)}-1=\tiny{\begin{pmatrix}1&-1\\\
0&1\end{pmatrix}}(w^{(i)})&\mbox{if $w^{(i)}\geq 1$},\\\
\dfrac{w^{(i)}}{1-w^{(i)}}=\tiny{\begin{pmatrix}1&0\\\
-1&1\end{pmatrix}}(w^{(i)})&\mbox{otherwise}.\end{array}\right.$

This algorithm is cyclic because the continued fraction of $w$ is purely
periodic; we denote by $\ell=\ell_{w}$ the length of the cycle, so that
$w^{(\ell+1)}=w^{(1)}$. The cycle $w^{(1)},\ldots,w^{(\ell)}$ corresponds to
the cycle of simple forms in the $\mathrm{SL}(2,\mathbb{Z})$-equivalence class
of the quadratic form $[a,b,c]$.

In [2] the value $j(w)$ is written in terms of the cycle (5.2) when $w$ is a
Markov irrationality (see [2, Lemma 4.1]) and in [3] for arbitrary quadratic
irrationalities. More concretely, in terms of simple forms, we have:

######  Lemma 5.1.

For any real quadratic irrationality $w$,

(5.3) $j(w)=\dfrac{1}{2\log\varepsilon}\int_{e^{\pi i/3}}^{e^{2\pi
i/3}}j(z)\sum_{\begin{subarray}{c}\left[a,b,c\right]\ \mathrm{simple}\\\
\left[a,b,c\right]\in\mathcal{A}\end{subarray}}\dfrac{\sqrt{D}}{az^{2}+bz+c}dz,$

where $\mathcal{A}$ is the $\mathrm{SL}(2,\mathbb{Z})$-equivalence class of
$w$ and $D$ its discriminant.

If $w$ has a purely periodic ‘–’ continued fraction expansion, we can write
(5.3) as:

(5.4)
$j(w)=\dfrac{1}{2\log\varepsilon}\int_{\pi/3}^{2\pi/3}j(e^{i\theta})ie^{i\theta}\sum_{i=1}^{\ell_{w}}\Big{(}\dfrac{1}{e^{i\theta}-w^{(i)}}-\dfrac{1}{e^{i\theta}-\tilde{w}^{(i)}}\Big{)}d\theta,$

where $w^{(i)}$ are defined in (5.2) and $\tilde{w}^{(i)}$ are the Galois
conjugates of $w^{(i)}$. Each term $w^{(i)}$ is of the form

$w^{(i)}=(a_{0},\overline{a_{1},\ldots,a_{n}})=a_{0}-\dfrac{1}{(\overline{a_{1},\ldots,a_{n}})}\qquad(1\leq
a_{0}\leq a_{n}-1).$

It is a well known fact (see for example [22]) that the conjugate of
$1/(\overline{a_{1},\ldots,a_{n}})$ is $(\overline{a_{n},\ldots,a_{1}})$ and
hence

$\tilde{w}^{(i)}=-(a_{n}-a_{0},\overline{a_{n-1},a_{n-2},\ldots,a_{1},a_{n}}).$

If $w$ is a Markov irrationality, the structure of the Markov-Hurwitz tree
implies that:

(5.5) $\displaystyle\dfrac{3}{8}=(1,2,3,2)\leq$ $\displaystyle
w^{(i)}\leq(3,2,4,2)=\dfrac{29}{12},$ (5.6)
$\displaystyle-\dfrac{21}{8}=-(3,3,3)\leq$
$\displaystyle\tilde{w}^{(i)}\leq-(1,2,3)=-\dfrac{2}{5}.$

##  6\. Stratregy for the proof of Theorem 1

Let $w$ be a Markov irrationality on the $n$-th level of the Markov-Hurwitz
tree, the first level corresponding to the Markov irrationality
$(\overline{2,3,4})$. Let $v$ be the ‘immediate’ predecessor of $w$, which is
on the same branch as $w$. Then we have one of the following two
configurations:

[$v$ [$w$] []] [$v$ [] [$w$]]

Let $u=(\overline{a_{1},\ldots,a_{s}})$ be the other predecessor of $w$ as
explained in section 3.1. In the second configuration, $u$ is the right
predecessor of $w$ and $v$. Then let
$(\overline{b})=(\overline{b_{1},\ldots,b_{t}})$ be the left predecessor of
$u$. We have $v=(\overline{u_{r},b})$ for some $r\geq 1$ and
$w=(\overline{u_{r+1},b})$. In the first configuration, $u$ is the left
predecessor of $w$ and $v$. If $w$ is not on the most left branch of the tree,
then we set $(\overline{b})=(\overline{b_{1},\ldots,b_{t}})$ be the right
predecessor of $u$, so that $v=(\overline{b,u_{r}})$ for some $r\geq 1$ and
$w=(\overline{b,u_{r+1}})$. The cycles are the same if we consider
$(\overline{u_{r},b})$ and $(\overline{u_{r+1},b})$, and we will refer to
these two situations as ‘case 1’. Note that in this case, $rs>n$.

The situation where $w$ is on the most left branch of the tree will be ‘case
2’. In this case, $u=(\overline{3})$, $v=(\overline{2,3_{r},4})$ for some
$r\geq 1$ and $w=(\overline{2,3_{r+1},4})$. Note that in this case, $r=n-1$
and

$\ell_{u}=2,\quad\ell_{v}=2r+4,\quad\ell_{w}=\ell_{u}+\ell_{v}=2(r+1)+4.$

Let

$J(w):=(2\log\varepsilon)j(w).$

In the next section we will write $J(w)$ as

(6.1) $J(w)=J(u)+J(v)+\delta_{w},$

where $\delta_{w}$ is the error term that we will explicitly bound in terms of
$n$. We see the identity (6.1) as a ‘local’ formula for $J(w)$ on the Markov-
Hurwitz tree. We will deduce a recursive formula for $J(w)$ and $q_{n}$ in
section 8 (see (8.7), (8.9), (8.10), (8.11)) that we see as a ‘global’ formula
and that will give the global interlacing property in section 9.

In order to obtain the local formula (6.1) we will compare the cycle of $w$ in
the algorithm (5.2) with the cycles of $u$ and $v$ and their $J$ values in
section 7.1. In sections 7.2, 7.3, 7.4 we bound the error term $\delta_{w}$.

##  7\. Local formula for $(2\log\varepsilon)j(w)$

###  7.1. Comparing cycles of $w,u,v$.

We will compare the cycle of $w$ in the algorithm (5.2) with the cycles of $u$
and $v$. We define a sum $S_{u}$ and a sum $S_{v}$ that contain all the terms
in the cycle of $u$ and $v$ respectively. The definitions of $S_{u}$ and
$S_{v}$ are different for case 1 and case 2. In case 1, the sum $S_{u}$ below
compares the first terms $w^{(1)},\ldots,w^{(\ell_{u})}$ and the conjugates
$\tilde{w}^{(r\ell u+1)},\ldots,\tilde{w}^{((r+1)\ell_{u})}$ with the cycle of
$u$ and conjugates, while the sum $S_{v}$ compares the remaining terms with
the cycle of $v$ and conjugates.

Case 1:

$\displaystyle
S_{u}(\theta)=\sum_{i=1}^{\ell_{u}}\dfrac{1}{e^{i\theta}-w^{(i)}}-\dfrac{1}{e^{i\theta}-u^{(i)}}+\sum_{i=1}^{\ell_{u}}\dfrac{1}{e^{i\theta}-\tilde{u}^{(i)}}-\dfrac{1}{e^{i\theta}-\tilde{w}^{(r\ell_{u}+i)}};$
$\displaystyle
S_{v}(\theta)=\sum_{i=1}^{\ell_{v}}\dfrac{1}{e^{i\theta}-w^{(i+\ell_{u})}}-\dfrac{1}{e^{i\theta}-v^{(i)}}+\sum_{i=1}^{r\ell_{u}}\dfrac{1}{e^{i\theta}-\tilde{v}^{(i)}}-\dfrac{1}{e^{i\theta}-\tilde{w}^{(i)}}$
$\displaystyle+\sum_{i=r\ell_{u}+1}^{\ell_{v}}\dfrac{1}{e^{i\theta}-\tilde{v}^{(i)}}-\dfrac{1}{e^{i\theta}-\tilde{w}^{(\ell_{u}+i)}}.$

Case 2:

$\displaystyle
S_{u}(\theta)=\sum_{i=1}^{2}\dfrac{1}{e^{i\theta}-w^{(1+i)}}-\dfrac{1}{e^{i\theta}-u^{(i)}}+\sum_{i=1}^{2}\dfrac{1}{e^{i\theta}-\tilde{u}^{(i)}}-\dfrac{1}{e^{i\theta}-\tilde{w}^{(1+2r+i)}};$
$\displaystyle
S_{v}(\theta)=\dfrac{1}{e^{i\theta}-w^{(1)}}-\dfrac{1}{e^{i\theta}-v^{(1)}}+\sum_{i=2}^{4+2r}\dfrac{1}{e^{i\theta}-w^{(2+i)}}-\dfrac{1}{e^{i\theta}-v^{(i)}}$
$\displaystyle+\sum_{i=1}^{1+2r}\dfrac{1}{e^{i\theta}-\tilde{v}^{(i)}}-\dfrac{1}{e^{i\theta}-\tilde{w}^{(i)}}+\sum_{i=2+2r}^{4+2r}\dfrac{1}{e^{i\theta}-\tilde{v}^{(i)}}-\dfrac{1}{e^{i\theta}-\tilde{w}^{(2+i)}}.$

Using (5.4), we have that

(7.1)
$J(w)-J(u)-J(v)=\int_{\pi/3}^{2\pi/3}j(e^{i\theta})ie^{i\theta}(S_{u}(\theta)+S_{v}(\theta))d\theta.$

Our next goal is to bound the integral in (7.1); we will give bounds for the
real and imaginary parts. For $\alpha\in\left\\{u,v\right\\}$, let

$\displaystyle\varepsilon_{\alpha}(\theta)$
$\displaystyle=\cos\theta\mathrm{Im}(S_{\alpha}(\theta))+\sin\theta\mathrm{Re}(S_{\alpha}(\theta));$
$\displaystyle\varepsilon^{\prime}_{\alpha}(\theta)$
$\displaystyle=\cos\theta\mathrm{Re}(S_{\alpha}(\theta))-\sin\theta\mathrm{Im}(S_{\alpha}(\theta)).$

The real and imaginary parts of the integral in (7.1) are respectively

(7.2)
$\mathrm{Re}(\eqref{intwithsums})=-\int_{\pi/3}^{2\pi/3}j(e^{i\theta})(\varepsilon_{u}(\theta)+\varepsilon_{v}(\theta))d\theta$

and

(7.3)
$\mathrm{Im}(\eqref{intwithsums})=\int_{\pi/3}^{2\pi/3}j(e^{i\theta})(\varepsilon^{\prime}_{u}(\theta)+\varepsilon^{\prime}_{v}(\theta))d\theta.$

Our goal is to bound (7.2) and (7.3). It will be useful to define the
following real functions with three variables
$(x,y,\theta)\in\mathbb{R}^{2}\times[\frac{\pi}{3},\frac{2\pi}{3}]$:

$g(x,y,\theta)=\dfrac{1}{x-y}\Big{(}\cos\theta\mathrm{Im}\Big{(}\dfrac{1}{e^{i\theta}-x}-\dfrac{1}{e^{i\theta}-y}\Big{)}+\sin\theta\mathrm{Re}\Big{(}\dfrac{1}{e^{i\theta}-x}-\dfrac{1}{e^{i\theta}-y}\Big{)}\Big{)},$
$g^{\prime}(x,y,\theta)=\dfrac{1}{x-y}\Big{(}\cos\theta\mathrm{Re}\Big{(}\dfrac{1}{e^{i\theta}-x}-\dfrac{1}{e^{i\theta}-y}\Big{)}-\sin\theta\mathrm{Im}\Big{(}\dfrac{1}{e^{i\theta}-x}-\dfrac{1}{e^{i\theta}-y}\Big{)}\Big{)}.$

Note that

(7.4)
$\dfrac{1}{e^{i\theta}-x}-\dfrac{1}{e^{i\theta}-y}=\dfrac{(x-y)(\mathrm{Re}((e^{i\theta}-x)(e^{i\theta}-y))-i\mathrm{Im}((e^{i\theta}-x)(e^{i\theta}-y))}{|(e^{i\theta}-x)(e^{i\theta}-y)|^{2}}.$

We have that

$\displaystyle\mathrm{Re}((e^{i\theta}-x)(e^{i\theta}-y))=\cos^{2}\theta-\sin^{2}\theta-\cos\theta(x+y)+xy,$
$\displaystyle\mathrm{Im}((e^{i\theta}-x)(e^{i\theta}-y))=2\sin\theta\cos\theta-\sin\theta(x+y),$

so

(7.5) $\displaystyle g(x,y,\theta)$
$\displaystyle=-\dfrac{\sin\theta(1-xy)}{((\cos\theta-x)^{2}+\sin^{2}\theta)((\cos\theta-y)^{2}+\sin^{2}\theta)}$
(7.6) $\displaystyle g^{\prime}(x,y,\theta)$
$\displaystyle=\dfrac{-x-y+\cos\theta(1+xy)}{((\cos\theta-x)^{2}+\sin^{2}\theta)((\cos\theta-y)^{2}+\sin^{2}\theta)}.$

For $x,y$ satisfying (5.5), one can easily compute:

(7.7) $\displaystyle-1.26964\leq g(x,y,\theta)\leq 0.354112,\quad-1.10636\leq
g^{\prime}(x,y,\theta)\leq-0.07222.$

For $x,y$ satisfying (5.6),

(7.8) $\displaystyle-1.25946\leq g(x,y,\theta)\leq 0.354112,\quad 0.04705\leq
g^{\prime}(x,y,\theta)\leq 1.10636.$

The following lemma (see [2, Lemma 2.1] for a proof) will also be useful:

######  Lemma 7.1.

If the ‘–’ continued fraction expansions of two Markov quadratics $u$ and $v$
coincide in the first $r$ partial quotients, then

$|u-v|\leq 10\Big{(}\dfrac{2}{1+\sqrt{5}}\Big{)}^{2(r-1)}.$

Let

$b(x)=10\Big{(}\dfrac{2}{1+\sqrt{5}}\Big{)}^{2(x-1)}$

be the bound of Lemma 7.1. Note that

(7.9) $\sum_{k=k_{0}}^{\infty}b(k)\leq
10\Big{(}\dfrac{2}{1+\sqrt{5}}\Big{)}^{2k_{0}-3}.$

###  7.2. Bounds for $\varepsilon_{u},\varepsilon^{\prime}_{u}$.

Case 1. We write $\varepsilon_{u}$ using the function $g(x,y,\theta)$:

$\varepsilon_{u}(\theta)=\sum_{i=1}^{\ell_{u}}(w^{(i)}-u^{(i)})g(w^{(i)},u^{(i)},\theta)+\sum_{i=1}^{\ell_{u}}(\tilde{u}^{(i)}-\tilde{w}^{(r\ell_{u}+i)})g(\tilde{u}^{(i)},\tilde{w}^{(r\ell_{u}+i)},\theta).$

We have a similar expression for $\varepsilon^{\prime}_{u}(\theta)$ replacing
$g$ by $g^{\prime}$. We apply Lemma 7.1 to bound $|w^{(i)}-u^{(i)}|$ and
$|\tilde{u}^{(i)}-\tilde{w}^{(r\ell_{u}+i)}|$ and use (7.7), (7.8) to bound
the rest.

In the first sum of $\varepsilon_{u}(\theta)$, the first $a_{1}-1$ terms
$w^{(i)}$ and $u^{(i)}$ share the same first $rs+s$ partial quotients, the
next $a_{2}-1$ terms share the first $rs+s-1$ partial quotients, etc. and the
last $a_{s}-1$ terms coincide in at least the first $rs+1$ partial quotients.

In the second sum $\varepsilon_{u}(\theta)$, we have a similar reversed
situation, where the first $a_{1}-1$ terms $\tilde{w}^{(r\ell_{u}+i)}$ and
$\tilde{u}^{(i)}$ share the same first $rs+1$ partial quotients, the next
$a_{2}-1$ terms share the same $rs+2$ first partial quotients, etc. and the
last $a_{s}-1$ terms share the same $rs+s$ partial quotients.

Using Lemma 7.1, (7.7), (7.8) and that $a_{i}-1\leq 3$ for $i=1,\ldots,s$, we
have

$\displaystyle|\varepsilon_{u}(\theta)|\leq
3(1.26964+1.25946)\,\sum_{k=1}^{s}b(rs+k)$
$\displaystyle|\varepsilon^{\prime}_{u}(\theta)|\leq 1.10636\cdot
6\sum_{k=1}^{s}b(rs+k).$

Case 2. We proceed analogously to case 1. The two terms in each sum of $S_{u}$
share the same first $r+1$ partial quotients. By Lemma 7.1 we have, for
$i=1,2$,

(7.10) $|w^{(1+i)}-u^{(i)}|,|\tilde{w}^{(i+1+2r)}-\tilde{u}^{(i)}|\leq
b(r+1).$

Using (7.7), (7.8) and (7.10), we have that

$\displaystyle|\varepsilon_{u}(\theta)|\leq 2\cdot 2.5291\,b(r+1)$
$\displaystyle|\varepsilon^{\prime}_{u}(\theta)|\leq 4\cdot 1.10636\,b(r+1).$

###  7.3. Bounds for $\varepsilon_{v},\varepsilon_{v}^{\prime}$.

We bound $\varepsilon_{v},\varepsilon_{v}^{\prime}$ in a similar way to
$\varepsilon_{u},\varepsilon_{u}^{\prime}$.

Case 1. In the first sum of $S_{v}$, the first $a_{1}-1$ terms
$w^{(\ell_{u}+i)}$ and $v^{(i)}$ coincide in the first $2rs+t$ partial
quotients, the next $a_{2}-1$ terms share the same first $2rs+t-1$ partial
quotients, etc. and the last $b_{t}-1$ terms coincide in the first $rs+1$
partial quotients.

In the second sum of $S_{v}$, the first $a_{1}-1$ terms $\tilde{w}^{(i)}$ and
$\tilde{v}^{(i)}$ share the same first $rs+t+1$ partial quotients, the next
$a_{2}-1$ terms coincide in the first $rs+t+2$ partial quotients, etc. and the
last $a_{s}-1$ terms coincide in the first $2rs+t$ partial quotients.

In the third sum of $S_{v}$, the first $b_{1}-1$ terms $\tilde{v}^{(i)}$ and
$\tilde{w}^{(\ell_{u}+i)}$ coincide in the first $rs+1$ partial quotients, the
next $b_{2}-1$ terms coincide in the first $rs+2$ partial quotients, etc. and
the last $b_{t}-1$ partial quotients coincide in the first $rs+t$ partial
quotients.

Using again Lemma 7.1, (7.7), (7.8) and that $a_{i}-1\leq 3$, $b_{j}-1\leq 3$
for $i\in\left\\{1,\ldots,s\right\\},j\in\left\\{1,\ldots,t\right\\}$, we have

$\displaystyle|\varepsilon_{v}(\theta)|\leq 3\cdot
2.5291\,\sum_{k=1}^{rs+t}b(rs+k)$
$\displaystyle|\varepsilon^{\prime}_{v}(\theta)|\leq 1.10636\cdot
6\sum_{k=1}^{rs+t}b(rs+k).$

Case 2. The terms $w_{1}$ and $v_{1}$ share the first same $r+1$ partial
quotients. In the first sum of $S_{v}$, the first two terms coincide in $2r+2$
partial quotients, each next two terms coincide in $2r+1,\ldots,r+3$ partial
quotients, and the last three terms coincide in $2+r$ partial quotients. In
the second sum, the first terms share the same $1+r$ first partial quotients,
and each next block of two terms $\tilde{w}_{i},\tilde{v}_{i}$ coincide in
$2+r$, $3+r$, etc.. $2+2r$ first partial quotients. In the third sum, the
terms coincide in $r+1$ partial quotients.

Using Lemma 7.1, (7.7), (7.8), we have that

$\displaystyle|\varepsilon_{v}(\theta)|\leq 1.26964(b(r+1)+b(r+2))+2.5291\cdot
2\sum_{k=2}^{r+2}b(r+k)+1.25946\cdot 4b(r+1)$
$\displaystyle|\varepsilon^{\prime}_{v}(\theta)|\leq
1.10636\Big{(}5b(r+1)+b(r+2)+4\sum_{k=2}^{r+2}b(r+k)\Big{)}.$

###  7.4. Conclusion.

Recall that we denote by $n$ the level on the tree where the Markov
irrationality $w$ lies. In case 1, using that $rs>n$, we have

(7.11) $\displaystyle|\varepsilon_{u}(\theta)|+|\varepsilon_{v}(\theta)|$
$\displaystyle\leq 151.7460\Big{(}\dfrac{2}{1+\sqrt{5}}\Big{)}^{2n+1},$ (7.12)
$\displaystyle|\varepsilon^{\prime}_{u}(\theta)|+|\varepsilon^{\prime}_{v}(\theta)|$
$\displaystyle\leq 132.7632\Big{(}\dfrac{2}{1+\sqrt{5}}\Big{)}^{2n+1}.$

In case 2, using that $r=n-1$, we have

(7.13) $\displaystyle|\varepsilon_{u}(\theta)|+|\varepsilon_{v}(\theta)|$
$\displaystyle\leq
113.6564\Big{(}\dfrac{2}{1+\sqrt{5}}\Big{)}^{2(n-1)}+63.278\Big{(}\dfrac{2}{1+\sqrt{5}}\Big{)}^{2n-1},$
(7.14)
$\displaystyle|\varepsilon^{\prime}_{u}(\theta)|+|\varepsilon^{\prime}_{v}(\theta)|$
$\displaystyle\leq
99.5724\Big{(}\dfrac{2}{1+\sqrt{5}}\Big{)}^{2(n-1)}+55.318\Big{(}\dfrac{2}{1+\sqrt{5}}\Big{)}^{2n-1}.$

Using (7.11), (7.12), (7.13), (7.14) and that

$\int_{\pi/3}^{2\pi/3}j(e^{i\theta})d\theta=753.982,$

we have

(7.15) $J(w)=J(u)+J(v)+\delta_{w},$

with

(7.16) $\displaystyle|\mathrm{Re}(\delta_{w})|\leq
115181.57371\Big{(}\dfrac{2}{1+\sqrt{5}}\Big{)}^{2(n-1)}$ (7.17)
$\displaystyle|\mathrm{Im}(\delta_{w})|\leq
100853.23866\Big{(}\dfrac{2}{1+\sqrt{5}}\Big{)}^{2(n-1)}.$

##  8\. Global formula for $(2\log\varepsilon)j(w)$

Consider a path on the Markov-Hurwitz tree given by Markov irrationalities
$w_{0}$, $w_{1}$, $w_{2}$, $\ldots,w_{n}$, $n\geq 2$. The quadratic $w_{n}$ is
on the $n$-th level of the tree, the first level corresponding to
$w_{1}=(\overline{2,3,4})$. If $w_{n}$ is on the left half of the tree,
$w_{0}=(\overline{3})$, otherwise $w_{0}=(\overline{2,4})$. Let
$r_{1},\ldots,r_{m}$ be the successive levels where the path turns, starting
from $r_{1}=1$, and let $r_{m+1}=n$. For convenience we set $r_{-1}=1$. We
write $q_{i}$ for the denominator of the Farey fraction associated to $w_{i}$.
If $w_{n}$ is on the left half of the tree, then $q_{0}=1$, otherwise
$q_{0}=2$. We write $\delta_{i}=\delta_{w_{i}}$ for the error in (7.15)
associated to $w_{i}$.

It follows from the relation (7.15) applied to $J(w_{n})$, that

(8.1) $J(w_{n})=J(w_{n-1})+J(w_{r_{m}-1})+\delta_{n}.$

By the construction of the Farey tree, we also have

(8.2) $q_{n}=q_{n-1}+q_{r_{m}-1}.$

If we apply (7.15) recursively to $J(w_{n}),\dots,J(w_{r_{m}})$ and
$q_{n},\ldots,q_{r_{m}}$ we have, if $m\geq 2$,

(8.3)
$J(w_{n})=(n-r_{m}+1)J(w_{r_{m}-1})+J(w_{r_{m-1}-1})+\sum_{i=r_{m}}^{n}\delta_{i},$
(8.4) $q_{n}=(n-r_{m}+1)q_{r_{m}-1}+q_{r_{m-1}-1}.$

If we apply again (7.15) recurrently to $J(w_{r_{k}-1}),\dots,J(w_{r_{k-1}})$
and $q_{r_{k}-1},\ldots,q_{r_{k-1}}$ we obtain, for $3\leq k\leq m$:

(8.5)
$J(w_{r_{k}-1})=(r_{k}-r_{k-1})J(w_{r_{k-1}-1})+J(w_{r_{k-2}-1})+\sum_{i=r_{k-1}}^{r_{k}-1}\delta_{i},$
(8.6) $q_{k}=(r_{k}-r_{k-1})q_{r_{k-1}-1}+q_{r_{k-2}-1}.$

Let $k_{0}$ be a positive constant ‘large enough’ that will indicate a level
‘down enough’ on the Markov-Hurwitz tree. We will choose a specific value for
$k_{0}$ later. We denote by $k$ and $s$ the positive integers such that

$k_{0}=r_{k-1}+s,\qquad 1\leq s\leq r_{k}-r_{k-1}.$

The recursive formulas (8.5),(8.3) give, if $m\geq 2$,

(8.7) $\begin{pmatrix}J(w_{n})\\\
J(w_{r_{m}-1})\end{pmatrix}=\begin{pmatrix}n-r_{m}+1&1\\\
1&0\end{pmatrix}\Big{(}\prod_{i={k+1}}^{m}\begin{pmatrix}r_{i}-r_{i-1}&1\\\
1&0\end{pmatrix}\Big{)}\begin{pmatrix}s&1\\\
1&0\end{pmatrix}\begin{pmatrix}J(w_{r_{k-1}-1})\\\
J(w_{r_{k-2}-1})\end{pmatrix}+\begin{pmatrix}\delta\\\
\delta^{\prime}\end{pmatrix}$

where

(8.8)
$\delta=\sum_{i=r_{m}}^{n}\delta_{i}+\Big{(}\sum_{j=k+1}^{m}\lambda_{j}\sum_{i=r_{j-1}}^{r_{j}-1}\delta_{i}\Big{)}+\lambda_{k}\sum_{i=k_{0}}^{r_{k}-1}\delta_{i}$

and $\lambda_{j}$ is the coefficient of $J(w_{r_{j-1}})$ in the recursive
formula (8.7) for $k\leq j\leq m$.

If $m=1$, then

(8.9)
$J(w_{n})=(n-k_{0})J(w_{0})+J(w_{k_{0}})+\sum_{i=k_{0}-1}^{n}\delta_{i}.$

We obtain similar formulas for $q_{n}$ from (8.6),(8.4): if $m\geq 2$,

(8.10) $\begin{pmatrix}q_{n}\\\
q_{r_{m}-1}\end{pmatrix}=\begin{pmatrix}n-r_{m}+1&1\\\
1&0\end{pmatrix}\Big{(}\prod_{i={k+1}}^{m}\begin{pmatrix}r_{i}-r_{i-1}&1\\\
1&0\end{pmatrix}\Big{)}\begin{pmatrix}s&1\\\
1&0\end{pmatrix}\begin{pmatrix}q_{r_{k-1}-1}\\\ q_{r_{k-2}-1}\end{pmatrix}$

and if $m=1$,

(8.11) $q_{n}=(n-k_{0})q_{0}+q_{k_{0}}.$

The coefficients of $q_{r_{j-1}}$ in (8.10) are the same as the coefficients
$\lambda_{j}$ for $J(w_{r_{j-1}})$ in (8.7), so we have, for all $j\geq 2$,

(8.12) $q_{n}\geq\lambda_{j}q_{r_{j-1}}$

and

(8.13) $\dfrac{J(w_{n})}{q_{n}}=\dfrac{\alpha J(w_{r_{k-2}-1})+\beta
J(w_{r_{k-1}-1})}{\alpha q_{r_{k-2}-1}+\beta
q_{r_{k-1}-1}}+\dfrac{\delta}{q_{n}},$

where $\alpha,\beta$ are some positive constants given by (8.7), (8.10) and
$\delta$ is given by (8.8).

##  9\. Asymptotic interlacing property

Suppose $n>k_{0}$. Below we bound $\delta/q_{n}$. We consider first the case
when $m\geq 2$. Using (7.16), (8.12) and $q_{r_{k-1}}\geq 4$, we have that

$\displaystyle\dfrac{\lambda_{k}}{q_{n}}\sum_{i=k_{0}}^{r_{k}-1}|\mathrm{Re}(\delta_{i})|$
$\displaystyle\leq\dfrac{115181.57371}{4}\sum_{i=k_{0}}^{r_{k}-1}\Big{(}\dfrac{2}{1+\sqrt{5}}\Big{)}^{2(i-1)}$
(9.1)
$\displaystyle\leq\dfrac{115181.57371}{4}\Big{(}\dfrac{2}{1+\sqrt{5}}\Big{)}^{2k_{0}-3}.$

Similarly, for $j\geq k+1$, using (8.12) and $q_{r_{j}-1}\geq r_{j-1}\geq
k_{0}$,

$\displaystyle\dfrac{1}{q_{n}}\sum_{j=k+1}^{m}\lambda_{j}\sum_{i=r_{j-1}}^{r_{j}-1}|\mathrm{Re}(\delta_{i})|$
$\displaystyle\leq\sum_{j=k+1}^{m}\dfrac{1}{q_{r_{j-1}}}\sum_{i=r_{j-1}}^{r_{j}-1}|\mathrm{Re}(\delta_{i})|$
(9.2)
$\displaystyle\leq\dfrac{115181.57371}{k_{0}}\Big{(}\dfrac{2}{1+\sqrt{5}}\Big{)}^{2k_{0}-1}.$

By (7.16) and $q_{n}>k_{0}$, we also have that

(9.3)
$\dfrac{|\mathrm{Re}(\delta_{n})|}{q_{n}}\leq\dfrac{115181.57371}{k_{0}+1}\Big{(}\dfrac{2}{1+\sqrt{5}}\Big{)}^{2k_{0}}.$

If $m=1$, then $q_{n}\geq n+2\geq k_{0}+3$, so

(9.4)
$\dfrac{|\mathrm{Re}(\delta)|}{q_{n}}=\dfrac{1}{q_{n}}\sum_{i=k_{0}-1}^{n}|\mathrm{Re}(\delta_{i})|\leq\dfrac{115181.57371}{k_{0}+3}\Big{(}\dfrac{2}{1+\sqrt{5}}\Big{)}^{2k_{0}-5}.$

Similar bounds for the imaginary parts can be found by using (7.17). Thus we
have, letting $k_{0}\rightarrow\infty$ as $n\rightarrow\infty$,

(9.5) $\frac{\delta}{q_{n}}\rightarrow 0\qquad as\ n\rightarrow\infty.$

By (8.1) and (8.2) we have, if
$\frac{\mathrm{Re}(J(w_{r_{m}-1}))}{q_{r_{m}-1}}\leq\frac{\mathrm{Re}(J(w_{n-1}))}{q_{n-1}}$,

$\dfrac{\mathrm{Re}(J(w_{r_{m}-1}))}{q_{r_{m}-1}}\leq\dfrac{\mathrm{Re}(J(w_{n}))}{q_{n}}\leq\dfrac{\mathrm{Re}(J(w_{n-1}))}{q_{n-1}}\qquad
as\ n\rightarrow\infty,$

and otherwise the inequalities are reversed. The same inequalities for the
imaginary parts hold. By (4.2), we conclude that $j(w_{n})$ lies between
$j(w_{n-1})$ and $j(w_{r_{m}-1})$, and hence Theorem 1 is proved.

##  10\. Asymptotic bounds for $j(w)$

By (8.13), we have

(10.1) $\dfrac{\mathrm{Re}(J(w_{r_{k-1}-1}))}{q_{r_{k-1}-1}}\leq\dfrac{\alpha
J(w_{r_{k-2}-1})+\beta J(w_{r_{k-1}-1})}{\alpha q_{r_{k-2}-1}+\beta
q_{r_{k-1}-1}}\leq\dfrac{\mathrm{Re}(J(w_{r_{k-2}-1}))}{q_{r_{k-2}-1}}$

if
$\frac{\mathrm{Re}(J(w_{r_{k-2}-1}))}{q_{r_{k-2}-1}}\geq\frac{\mathrm{Re}(J(w_{r_{k-1}-1}))}{q_{r_{k-1}-1}}$,
and otherwise the inequalities are reversed in (10.1). We have similar bounds
for the imaginary parts.

Let us choose $k_{0}=12$. We checked computationally that for all $\ell\leq
12$,

(10.2) $\displaystyle 1251.36168$
$\displaystyle\leq\dfrac{\mathrm{Re}(J(w_{\ell}))}{q_{\ell}}\leq 1359.5674,$
(10.3) $\displaystyle-0.4813$
$\displaystyle\leq\dfrac{\mathrm{Im}(J(w_{\ell}))}{q_{\ell}}\leq 0.$

By (10.1),

$\displaystyle 1251.36168+\dfrac{\mathrm{Re}(\delta)}{q_{n}}$
$\displaystyle\leq\dfrac{\mathrm{Re}(J(w_{n}))}{q_{n}}\leq
1359.5674+\dfrac{\mathrm{Re}(\delta)}{q_{n}}$
$\displaystyle-0.4813+\dfrac{\mathrm{Im}(\delta)}{q_{n}}$
$\displaystyle\leq\dfrac{\mathrm{Im}(J(w_{n}))}{q_{n}}\leq\dfrac{\mathrm{Im}(\delta)}{q_{n}}.$

From the bounds (9), (9), (9.3), (9.4) with $k_{0}=12$ we obtain

(10.4) $\displaystyle\dfrac{|\mathrm{Re}(\delta)|}{q_{n}}$ $\displaystyle\leq
1.41173,$ (10.5) $\displaystyle\dfrac{|\mathrm{Im}(\delta)|}{q_{n}}$
$\displaystyle\leq 1.23611.$

By (4.2), (7.16) and (7.17) we have

(10.6) $\displaystyle 3206.24623\leq\dfrac{\mathrm{Re}(J(w_{n}))}{\sqrt{n}}$
$\displaystyle\leq 3491.04708\qquad(as\ n\rightarrow\infty),$ (10.7)
$\displaystyle-4.40533\leq\dfrac{\mathrm{Im}(J(w_{n}))}{\sqrt{n}}$
$\displaystyle\leq 3.170734\qquad(as\ n\rightarrow\infty)$

and by (4.5),

(10.8) $\displaystyle 681.50081\leq\mathrm{Re}(j(w_{n}))$ $\displaystyle\leq
742.03641\qquad(as\ n\rightarrow\infty)$ (10.9)
$\displaystyle-0.93637\leq\mathrm{Im}(j(w_{n}))$ $\displaystyle\leq
0.67396\qquad(as\ n\rightarrow\infty).$

######  Remark 10.1.

The condition $n\rightarrow\infty$ in (10.6), (10.7), (10.8) and (10.9) could
be replaced by $n\geq n_{0}$ for an explicit value of $n_{0}$ if the
asymptotic relations (4.2) and (4.3) are made explicit. Then one could compute
the values $J(w_{n})$, $j(w_{n})$ for $n<n_{0}$ and hence obtain upper and
lower bounds for all $n\geq 1$.

##  11\. Appendix

The first table below shows the values of $J(w(p/q)/q)$ and $j(w(p/q))$ for
the first 40 Farey fractions $p/q$ (ordered as real numbers) among all Farey
fractions up to the level $2^{12}=4096$ on the Farey tree. The second table
shows the last $40$ values. The programs used for the computations were done
in collaboration with Don Zagier.

$p/q$ | $J(w(p/q))/q$ | $j(w(p/q))$
---|---|---
0 | 1359.56741044 | 706.324813541
1/14 | 1341.67984291 - 0.122490502636*I | 706.858789119 - 0.0645336432753*I
1/13 | 1340.30387617 - 0.131912848914*I | 706.900488474 - 0.0695732206634*I
2/25 | 1339.53333481 - 0.137189362590*I | 706.923879686 - 0.0724001664859*I
1/12 | 1338.69858166 - 0.142905585739*I | 706.949252302 - 0.0754665750541*I
3/35 | 1338.10232944 - 0.146988601101*I | 706.967396097 - 0.0776593436020*I
2/23 | 1337.79124132 - 0.149118869986*I | 706.976869216 - 0.0788042174199*I
3/34 | 1337.47100355 - 0.151311793837*I | 706.986625824 - 0.0799833523780*I
1/11 | 1336.80141549 - 0.155896998255*I | 707.007041981 - 0.0824507472184*I
4/43 | 1336.27197388 - 0.159522502181*I | 707.023200307 - 0.0844035587194*I
3/32 | 1336.08997833 - 0.160768769155*I | 707.028757860 - 0.0850752157429*I
5/53 | 1335.94232156 - 0.161779891418*I | 707.033267994 - 0.0856202872524*I
2/21 | 1335.71732077 - 0.163320649151*I | 707.040142609 - 0.0864511175163*I
5/52 | 1335.48799304 - 0.164891036840*I | 707.047151949 - 0.0872982300007*I
3/31 | 1335.33264200 - 0.165954847856*I | 707.051901658 - 0.0878722552532*I
4/41 | 1335.13561141 - 0.167304071583*I | 707.057927362 - 0.0886004905281*I
1/10 | 1334.52481658 - 0.171486665136*I | 707.076619003 - 0.0908594653933*I
5/49 | 1334.01374406 - 0.174986346915*I | 707.092272845 - 0.0927512885887*I
4/39 | 1333.88269983 - 0.175883701217*I | 707.096288695 - 0.0932366184737*I
7/68 | 1333.78827089 - 0.176530324170*I | 707.099182986 - 0.0935864040170*I
3/29 | 1333.66128026 - 0.177399920555*I | 707.103075992 - 0.0940568878788*I
8/77 | 1333.54913268 - 0.178167875804*I | 707.106514624 - 0.0944724589373*I
5/48 | 1333.48137685 - 0.178631848767*I | 707.108592428 - 0.0947235689505*I
7/67 | 1333.40350822 - 0.179165071426*I | 707.110980627 - 0.0950121914106*I
2/19 | 1333.20678745 - 0.180512160249*I | 707.117015286 - 0.0957415017533*I
7/66 | 1333.00708607 - 0.181879659509*I | 707.123143307 - 0.0964820951663*I
5/47 | 1332.92635572 - 0.182432478358*I | 707.125621143 - 0.0967815506251*I
8/75 | 1332.85531302 - 0.182918958946*I | 707.127801902 - 0.0970451031725*I
3/28 | 1332.73606276 - 0.183735551361*I | 707.131463014 - 0.0974875617700*I
7/65 | 1332.59846631 - 0.184677773378*I | 707.135688236 - 0.0979981950158*I
4/37 | 1332.49433927 - 0.185390806256*I | 707.138886314 - 0.0983846943323*I
5/46 | 1332.34720323 - 0.186398352714*I | 707.143406240 - 0.0989309436276*I
1/9 | 1331.74231064 - 0.190540488152*I | 707.161999257 - 0.101177976749*I
6/53 | 1331.21731397 - 0.194135299692*I | 707.178150980 - 0.103129850277*I
5/44 | 1331.10992829 - 0.194870602053*I | 707.181456402 - 0.103529297799*I
9/79 | 1331.03788473 - 0.195363906168*I | 707.183674280 - 0.103797319799*I
4/35 | 1330.94731569 - 0.195984059913*I | 707.186462829 - 0.104134305293*I
11/96 | 1330.87278491 - 0.196494394766*I | 707.188757875 - 0.104411652668*I
7/61 | 1330.83002135 - 0.196787209845*I | 707.190074826 - 0.104570801246*I
10/87 | 1330.78283397 - 0.197110316139*I | 707.191528119 - 0.104746426028*I

$p/q$ | $J(w(p/q))/q$ | $j(w(p/q))$
---|---|---
67/144 | 1256.80214081 - 0.102528424246*I | 709.686203382 - 0.0578953566189*I
47/101 | 1256.79136764 - 0.102325397664*I | 709.686610778 - 0.0577812408124*I
74/159 | 1256.78161080 - 0.102141524533*I | 709.686979746 - 0.0576778888468*I
27/58 | 1256.76462045 - 0.101821331667*I | 709.687622275 - 0.0574979097853*I
61/131 | 1256.74399856 - 0.101432700632*I | 709.688402163 - 0.0572794549412*I
34/73 | 1256.72761405 - 0.101123925288*I | 709.689021820 - 0.0571058826255*I
41/88 | 1256.70322348 - 0.100664271084*I | 709.689944295 - 0.0568474876195*I
7/15 | 1256.58452267 - 0.0984272872898*I | 709.694434219 - 0.0555898124674*I
36/77 | 1256.44886461 - 0.0958707343734*I | 709.699566670 - 0.0541521589596*I
29/62 | 1256.41604412 - 0.0952522135066*I | 709.700808570 - 0.0538042897974*I
51/109 | 1256.39285899 - 0.0948152767474*I | 709.701685917 - 0.0535585356735*I
22/47 | 1256.36227436 - 0.0942388920864*I | 709.702843321 - 0.0532343345784*I
59/126 | 1256.33581623 - 0.0937402736098*I | 709.703844616 - 0.0529538613138*I
37/79 | 1256.32007532 - 0.0934436271744*I | 709.704440344 - 0.0527869914925*I
52/111 | 1256.30220725 - 0.0931068933828*I | 709.705116596 - 0.0525975662884*I
15/32 | 1256.25809547 - 0.0922755818348*I | 709.706786180 - 0.0521298981978*I
53/113 | 1256.21476442 - 0.0914589837656*I | 709.708426336 - 0.0516704732988*I
38/81 | 1256.19764598 - 0.0911363771209*I | 709.709074332 - 0.0514889627848*I
61/130 | 1256.18276611 - 0.0908559574990*I | 709.709637605 - 0.0513311839730*I
23/49 | 1256.15816877 - 0.0903924066955*I | 709.710568762 - 0.0510703571911*I
54/115 | 1256.13036308 - 0.0898683927437*I | 709.711621420 - 0.0507754964001*I
31/66 | 1256.10971946 - 0.0894793520826*I | 709.712402972 - 0.0505565755912*I
39/83 | 1256.08111686 - 0.0889403198412*I | 709.713485890 - 0.0502532388899*I
8/17 | 1255.97007146 - 0.0868476064334*I | 709.717690658 - 0.0490754390393*I
33/70 | 1255.83840334 - 0.0843662462496*I | 709.722677340 - 0.0476786169351*I
25/53 | 1255.79617017 - 0.0835703382661*I | 709.724277078 - 0.0472305134542*I
42/89 | 1255.76295307 - 0.0829443432229*I | 709.725535379 - 0.0468780499190*I
17/36 | 1255.71405012 - 0.0820227394093*I | 709.727388008 - 0.0463591090605*I
43/91 | 1255.66622196 - 0.0811213906246*I | 709.729200069 - 0.0458515317763*I
26/55 | 1255.63491625 - 0.0805314168746*I | 709.730386225 - 0.0455192770302*I
35/74 | 1255.59641869 - 0.0798059086145*I | 709.731844962 - 0.0451106692538*I
9/19 | 1255.48497839 - 0.0777057531246*I | 709.736068162 - 0.0439277065400*I
28/59 | 1255.34520582 - 0.0750716597984*I | 709.741366188 - 0.0424436737723*I
19/40 | 1255.27881384 - 0.0738204654684*I | 709.743883191 - 0.0417386346705*I
29/61 | 1255.21459865 - 0.0726102939033*I | 709.746317939 - 0.0410566358914*I
10/21 | 1255.09228401 - 0.0703052052080*I | 709.750956292 - 0.0397573845879*I
21/44 | 1254.92271143 - 0.0671095140622*I | 709.757388330 - 0.0379557027688*I
11/23 | 1254.76788430 - 0.0641917091029*I | 709.763262682 - 0.0363102351122*I
12/25 | 1254.49538854 - 0.0590563723747*I | 709.773605298 - 0.0334131593620*I
1/2 | 1251.36168734 | 709.892890920

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