# Diophantine approximation in metric space

Jonathan M. Fraser  Mathematical Institute, University of St Andrews,
Scotland, KY16 9SS<EMAIL_ADDRESS>,  Henna Koivusalo  School of
Mathematics, University of Bristol, BS8 1UG<EMAIL_ADDRESS>and
Felipe A. Ramírez  Weslyan University, Middletown, CT, USA
<EMAIL_ADDRESS>
###### Abstract.

Diophantine approximation is traditionally the study of how well real numbers
are approximated by rationals. We propose a model for studying Diophantine
approximation in an arbitrary totally bounded metric space where the rationals
are replaced with a countable hierarchy of “well-spread” points, which we
refer to as _abstract rationals_. We prove various Jarník–Besicovitch type
dimension bounds and investigate their sharpness.

The authors were supported by a Mathematisches Forschungsinstitut Oberwolfach
- Research in Pairs Grant. JMF was financially supported by an _EPSRC Standard
Grant_ (EP/R015104/1) and a _Leverhulme Trust Research Project Grant_
(RPG-2019-034).

##  1\. Introduction

###  1.1. Diophantine approximation in a metric space

Let $(F,d)$ be a non-empty totally bounded metric space. For $q\in\mathbb{N}$,
define _abstract rationals of level $q$_ by fixing a maximal $(1/q)$-separated
set $P(q)\subseteq F$. By $(1/q)$-_separated_ we mean that for all
$p_{1},p_{2}\in P(q)$, $d(p_{1},p_{2})\geqslant 1/q$ and by _maximal_ we mean
it is impossible to add another point to the set and maintain the separation.
Since $F$ is totally bounded, $P(q)$ is a finite set for all $q$. One should
think of $P(q)$ as the set of rationals with denominator $q$ inside $F$. We
refer to the set $\cup_{q}P(q)$ as a set of _abstract rationals_ , noting that
it is immediately a countable dense subset of $F$. For $t>0$ let

$\displaystyle F_{t}$ $\displaystyle=$ $\displaystyle\\{x\in F:\text{ i.o. for
some $p\in P(q)$ }\,d(p,x)\leqslant 1/q^{t}\\}$ $\displaystyle=$
$\displaystyle\bigcap_{N}\bigcup_{q>N}\bigcup_{p\in P(q)}B(p,1/q^{t})$

where $B(x,r)$ generally denotes the closed ball centred at $x\in X$ with
radius $r>0$. Clearly, if $t\leqslant 1$, then $F_{t}=F$ and so we are
interested in $t>1$. Computing (or estimating) the size (e.g. Hausdorff
dimension) of $F_{t}$ describes “how much of $F$ can be far away from a chosen
set of well spread out points”.

Classical Diophantine approximation may be cast in this language. For example,
if $F=[0,1]$ with the Euclidean distance and $P(q)=\mathbb{Z}/q\cap[0,1]$ are
the rationals with denominator $q$, then $F_{t}$ is the set of $t$-well-
approximable real numbers in $[0,1]$. In this setting, Dirichlet’s theorem
tells us that $F_{2}=[0,1]$, the Borel–Cantelli lemma tells us that $F_{t}$ is
a null set for all $t>2$, and the Jarník–Besicovitch theorem gives

$\dim_{\textup{H}}F_{t}=2/t$

for all $t\geqslant 2$, where $\dim_{\textup{H}}$ denotes the Hausdorff
dimension. That is, the exponent $t_{0}=2$ is a natural threshold beyond which
it is sensible to study the dimensions of $F_{t}$. Our aim here is to
establish analogous results in the general setting of abstract rationals.

The inspiration for studying abstract rationals comes from problems of
_intrinsic_ Diophantine approximation. In such problems, one has a subset $X$
of $\mathbb{R}^{n}$, and the goal is to understand how well its points can be
approximated by rational points which also lie on $X$. The set $X$ can for
example be taken to be a manifold [FKMS18] or a fractal set [BFR11]. Questions
on intrinsic Diophantine approximation on fractals have attracted much
attention and there have been many interpretations and answers to the now-
classical Mahler’s problem: ‘How well can irrationals in the Cantor set be
approximated by rationals in the Cantor set?’ Interpretations include random
rational points [BD16], intrinsic rationals [FMS18, FS14, LSV07, S21], and
dynamical rationals [B19, AB].

In Section 2 we provide analogues to the Jarník–Besicovitch theorem in metric
spaces using the language of abstract rationals. It turns out that the problem
is subtle and the answer depends on how the abstract rationals are chosen.
Despite the generality of the notion of abstract rationals, the bounds we
prove in this general set-up are sharp in many cases of particular interest.
For example, Theorem 2.1 below demonstrates that the upper bound for the
dimension value in the Jarník–Besicovitch theorem holds in much more
generality, replacing “$2$” with “$\dim_{\textup{P}}F+1$” where
$\dim_{\textup{P}}$ denotes the _packing_ dimension. We also show in
Proposition 2.2 that in general one cannot replace “$2$” with
“$\dim_{\textup{H}}F+1$” or “$2\dim_{\textup{H}}F$”, for example. In order to
estimate $\dim_{\textup{H}}F_{t}$ in general we naturally encounter several
different notions of fractal dimension, which we recall in the following
subsection.

We obtain stronger dimension bounds in the special case when $F$ is
Ahlfors–David regular, see Theorem 2.7. In Section 3, during the investigation
of the sharpness of the bounds in the Ahlfors–David regular case, we encounter
the notions of Dirichlet exponent and Dirichlet spectrum. By Dirichlet
exponent we mean the threshold $t_{0}$ after which $F_{t}$ does not
(essentially) contain a ball. The definition is a natural extension of the
classical Dirichlet exponent ($t_{0}=2$) to the setting of abstract rationals.
We find that in general the Dirichlet exponent of $F$ depends on the choice of
abstract rationals, so we are occasioned to consider the collection of
possible Dirichlet exponents for a given $F$—the Dirichlet spectrum of $F$.
The main result of Section 2 and the mass transference principle of
Beresnevich and Velani [BV06] allow us to bound the Dirichlet spectrum and
reveal an interplay between Dirichlet exponents and the dimensions of $F_{t}$.
In studying that interplay, we find that the dimension bounds from Theorem 2.7
are sharp.

###  1.2. Dimension theory

An important problem in the study of fractal objects is to characterise their
dimension. There are many ways to define dimension and the relationship
between the notions is especially interesting, and relevant to us. See [F14,
F20] for more background on dimension theory and fractal geometry.

Given $s\geqslant 0$, the _$s$ -dimensional Hausdorff measure_ of a set $X$ in
a metric space is defined by

$\mathcal{H}^{s}(X)=\lim_{r\to 0}\,\inf\bigg{\\{}\sum_{i}\lvert
U_{i}\rvert^{s}:\\{U_{i}\\}_{i}\text{ is a countable $r$-cover of
$X$}\bigg{\\}}.$

The _Hausdorff dimension_ of $X$ is then

$\dim_{\textup{H}}X=\inf\Big{\\{}s\geqslant
0:\mathcal{H}^{s}(X)=0\Big{\\}}=\sup\Big{\\{}s\geqslant
0:\mathcal{H}^{s}(X)=\infty\Big{\\}}.$

A subtle feature of the Hausdorff dimension is that sets of different sizes
may be used in the covers and their contribution is weighted accordingly. The
(upper) box dimension is a simpler notion of dimension which avoids this
subtlety. The _upper box dimension_ of $X$ is

$\overline{\dim}_{\textup{B}}X=\limsup_{r\to 0}\,\frac{\log N_{r}(X)}{-\log
r},$

where $N_{r}(X)$ is maximum cardinality of an $r$-separated subset of $X$. If
the $\limsup$ is really a limit then we refer to the box dimension, written
$\dim_{\text{B}}X$. Note that if $X$ is not totally bounded then $N_{r}(X)$
may be infinite.

The packing dimension of $X$ may be defined via packing measure in an
analogous way to the Hausdorff dimension, but we use a common alternative
formulation in terms of the upper box dimension. We define the _packing
dimension_ of $X$ by

$\dim_{\textup{P}}X=\inf\left\\{\sup_{i}\overline{\dim}_{\textup{B}}X_{i}\ :\
X=\bigcup_{i}X_{i}\right\\}.$

It should be clear from the definitions that

$\dim_{\textup{H}}X\leqslant\dim_{\textup{P}}X\leqslant\overline{\dim}_{\textup{B}}X.$

Finally, the _lower dimension_ of $X$ is defined by

$\displaystyle\dim_{\textup{L}}X\ =\ \sup\Bigg{\\{}\ s\geqslant 0$
$\displaystyle:$ there exists a constant $C>0$ such that, for all
$0<r<R\leqslant|X|$ and $x\in X$, $\displaystyle\hskip
76.82243pt\text{$N_{r}\big{(}B(x,R)\cap X\big{)}\ \geqslant\
C\bigg{(}\frac{R}{r}\bigg{)}^{s}$ }\Bigg{\\}}$

where $|X|$ denotes the diameter of $X$. The lower dimension is the natural
dual to the Assouad dimension and identifies the “thinnest part” of the space.
If $X$ is closed, then

$\dim_{\textup{L}}X\leqslant\dim_{\textup{H}}X.$

See [F20, Chapter 3] for more background on the lower dimension, including its
relevance in a different area of Diophantine approximation, e.g. [F20, Section
14.2].

##  2\. Main results

###  2.1. General bounds

Our first result provides general bounds for the dimension of $F_{t}$ which
hold independently of how the abstract rationals are chosen. The general lower
bound assumes compactness of $F$. If $F$ is not compact, then this lower bound
does not hold. For example, if $F=[0,1]\cap\mathbb{Q}$, then the packing
dimension of $F=F_{1}$ is $0$ and the lower dimension is $1$.

######  Theorem 2.1.

For all $t\geqslant 1$

$\dim_{\textup{H}}F_{t}\leqslant\frac{\dim_{\textup{P}}F+1}{t}$

and, if $F$ is compact,

$\dim_{\textup{P}}F_{t}\geqslant\frac{\dim_{\textup{L}}F}{t}.$

  * Proof.

We begin with the general upper bound. Let $b>\dim_{\textup{P}}F$ and let

$F=\bigcup_{i}F_{i}$

be a countable decomposition of $F$ with the property that
$\overline{\dim}_{\textup{B}}F_{i}<b$ for all $i$. Temporarily fix an index
$i$. We write

$d(p,F_{i})=\inf\\{d(p,x):x\in F_{i}\\}.$

For all $N\geqslant 1$

$\bigcup_{q>N}\bigcup_{\begin{subarray}{c}p\in P(q):\\\ d(p,F_{i})\leqslant
1/q\end{subarray}}\\{B(p,1/q^{t})\\}$

provides a $2/N^{t}$-cover of $F_{t}\cap F_{i}$. Therefore, for some constant
$C_{1}\geqslant 1$ depending on $F$, $i$ and $b$,

$\displaystyle\mathcal{H}^{s}_{2/N^{t}}(F_{t}\cap F_{i})$
$\displaystyle\leqslant$ $\displaystyle\sum_{q>N}\sum_{\begin{subarray}{c}p\in
P(q):\\\ d(p,F_{i})\leqslant 1/q\end{subarray}}|B(p,1/q^{t})|^{s}$
$\displaystyle=$ $\displaystyle\sum_{q>N}(\\#\\{p\in P(q):d(p,F_{i})\leqslant
1/q\\})(2/q^{t})^{s}$ $\displaystyle\leqslant$ $\displaystyle
2^{s}C_{1}\sum_{q\geqslant 1}q^{b-ts}<\infty$

provided $s>\frac{b+1}{t}$. For such $s$

$\mathcal{H}^{s}(F_{t})\leqslant\sum_{i}\mathcal{H}^{s}(F_{t}\cap F_{i})=0$

and the desired result follows.

Turning our attention to the general lower bound, let $0<l<\dim_{\textup{L}}F$
(we may assume the lower dimension is strictly positive). Fix a rapidly
increasing sequence of integers $q_{k}$ satisfying

(1) $q_{1}q_{2}\cdots q_{k}\leqslant\log(q_{k+1})$

and construct a nested decreasing sequence of compact sets $E_{k}\subseteq
F_{t}$ by setting $E_{1}=B(z,q_{1}^{-t})$ with $z\in F$ chosen arbitrarily and
then letting $E_{k}$ be the union of the closed balls $B(p,q_{k}^{-t})$ with
$p\in P(q_{k})$ which lie completely inside $E_{k-1}$. The sets $E_{k}$ are
compact since $F$ is compact and so by the Cantor intersection theorem

$E=\bigcap_{k}E_{k}\subseteq F_{t}$

with $E$ non-empty and compact. We show $\dim_{\textup{P}}E\geqslant l/t$,
which completes the proof. Let $\mu$ be a probability measure supported on $E$
constructed by assigning unit mass to $E_{1}$ and then, given a $k$-level
cylinder $B(p,q_{k}^{-t})$ used in the construction of $E_{k}$, redistribute
the mass of $B(p,q_{k}^{-t})$ equally among the $(k+1)$-level cylinders lying
inside. Since the $q_{k}$ are rapidly increasing and $P(q_{k})$ is
$(1/q_{k})$-separated, using the definition of lower dimension, we may assume
that there are at least

$\left(\frac{q_{k}^{-t}}{q_{k+1}^{-1}}\right)^{l}$

$(k+1)$-level cylinders inside every $k$-level cylinder. For each $k$,
consider $r=q_{k}^{-t}$. It follows that any ball $B(x,r)$ with $x\in E$
intersects at most one $k$-level cylinder. Recall that the centres of the
$k$-level cylinders are $(1/q_{k})$-separated. Therefore

$\displaystyle\mu(B(x,r))$ $\displaystyle\leqslant$
$\displaystyle\left(\frac{q_{1}^{-t}}{q_{2}^{-1}}\right)^{-l}\left(\frac{q_{2}^{-t}}{q_{3}^{-1}}\right)^{-l}\cdots\left(\frac{q_{k-1}^{-t}}{q_{k}^{-1}}\right)^{-l}$
$\displaystyle=$ $\displaystyle q_{1}^{tl}(q_{2}\cdots
q_{k-1})^{tl-l}(q_{k}^{-1})^{l}$ $\displaystyle\leqslant$
$\displaystyle\left(\log(q_{k})\right)^{tl}(q_{k}^{-1})^{l}\qquad\text{by
\eqref{rapid}}$ $\displaystyle=$ $\displaystyle\left(\frac{|\log
r|}{t}\right)^{tl}r^{l/t}.$

This proves that for all $x\in E$, the upper local dimension of $\mu$ at $x$
is at least $l/t$. Applying standard estimates for packing dimension coming
from local dimension of measures, e.g. [F97, Proposition 2.3], this proves
that $\dim_{\textup{P}}E\geqslant l/t$ as required. ∎

###  2.2. Sharpness of general bounds

####  2.2.1. Sharpness of upper bound

Comparing the upper bound from Theorem 2.1 with the classical
Jarník–Besicovitch theorem, one might ask if the packing dimension can be
replaced by the Hausdorff dimension, or indeed if an upper bound of the form
$c\dim_{\textup{H}}F/t$ is possible for some uniform constant $c$. The
following proposition shows that this is not possible and demonstrates that
the packing dimension should appear in bounding the Hausdorff dimension of
$F_{t}$.

######  Proposition 2.2.

For all $t>2$, there exists a compact set $F\subseteq[0,1]$ and a choice of
abstract rationals such that

$\dim_{\textup{P}}F=1$

and

$\dim_{\textup{H}}F_{t}=\dim_{\textup{H}}F=2/t=\frac{\dim_{\textup{P}}F+1}{t}.$

In particular we cannot replace the packing dimension by the Hausdorff
dimension in Theorem 2.1.

  * Proof.

Let $E_{t}$ denote the set of $t$-well-approximable numbers in $[0,1]$ in the
classical setting. Then

$\dim_{\textup{H}}E_{t}=2/t$

and $\mathcal{H}^{2/t}(E_{t})=\infty$. This is the Jarník–Besicovitch theorem,
but we emphasise the infinitude of the Hausdorff measure which we need for
what follows, see [BBDF09, Theorem 11]. By [F14, Theorem 4.10] there exists a
compact subset $E_{t}^{*}\subseteq E_{t}$ such that

$\dim_{\textup{H}}E_{t}^{*}=2/t.$

Let

$F=E_{t}^{*}\cup\\{p/q:p\in\mathbb{Z},\,q\in\mathbb{N},\,|p/q-x|\leqslant
1/q\text{ for some }x\in E_{t}^{*}\\}.$

Then $F$ is compact and

$\dim_{\textup{H}}F=\dim_{\textup{H}}E_{t}=2/t.$

The abstract rationals $P(q)$ are then chosen to be $P(q)=\mathbb{Z}/q\cap F$.
By construction $E_{t}^{*}\subseteq E_{t}\cap F=F_{t}$ and so

$\dim_{\textup{H}}F_{t}=\dim_{\textup{H}}F=2/t.$

Moreover, by Theorem 2.1,

$2/t=\dim_{\textup{H}}F_{t}\leqslant\frac{\dim_{\textup{P}}F+1}{t}$

which forces $\dim_{\textup{P}}F=1$, completing the proof. ∎

The next result is along similar lines but this time the construction works
for arbitrary choices of abstract rationals.

######  Proposition 2.3.

For all integers $t\geqslant 2$, there exists a compact set $F\subseteq[0,1]$
such that

$F_{s}=F$

for all $1\leqslant s<t$ and all choices of abstract rationals
$\mathcal{P}=\bigcup_{q}P(q)$. In particular, for all $t\geqslant 1$,

$\sup_{F}\inf_{\mathcal{P}}\frac{\dim_{\textup{H}}F_{t}}{\dim_{\textup{H}}F}=1$

where the supremum is taken over compact sets $F\subseteq[0,1]$ with
$\dim_{\textup{H}}F>0$ and the infimum is taken over all choices of abstract
rationals for $F$.

  * Proof.

Consider an iterated function system (IFS) type construction which alternates
between $\\{x\mapsto x/2,x\mapsto(x+1)/2\\}$ and $\\{x\mapsto x/2\\}$. See
[F14] for background on IFS theory. Let $N(k)$ be the number of times we
choose the first IFS in the first $k$ levels. Then it is standard that

$\dim_{\textup{H}}F=\liminf_{k\to\infty}N(k)/k$

and

$\dim_{\textup{P}}F=\limsup_{k\to\infty}N(k)/k.$

Let $k_{n}$ be a very rapidly increasing sequence of integers, e.g.
$k_{n}=n^{n}$. Fix an integer $t\geqslant 2$ and construct $F$ inductively by
using the first IFS $k_{n}$ times, followed by the second IFS $(t-1)k_{n}$
times. This is repeated in blocks of $tk_{n}$. Then

$\dim_{\textup{H}}F=1/t<1=\dim_{\textup{P}}F.$

Consider $x\in F$ and especially the cylinder $E$ of level

$M=t\sum_{n=1}^{N}k_{n}$

containing $x$. (Just after the $N$th long run of the second IFS). This
cylinder has size $2^{-M}\leqslant 2^{-tk_{N}}$. There are very large holes to
the left and right of this cylinder, of size

$>2\cdot 2^{-k_{N}\Delta}$

where $\Delta>1$ can be chosen arbitrarily provided $N$ is large enough. Due
to the presence of these holes, there must exist $p\in P(q)\cap E$ for some
$q$ with $1/q\geqslant 2^{-k_{N}\Delta}$. This ensures that the whole set $F$
is contained in the

$2^{-tk_{N}}=\left(2^{-k_{N}\Delta}\right)^{t/\Delta}\leqslant(1/q)^{t/\Delta}$

neighbourhood of $P(q)$ for infinitely many $q$. This ensures that

$F_{t/\Delta}=F$

for all $\Delta>1$. ∎

####  2.2.2. Sharpness of lower bound

Comparing the lower bound from Theorem 2.1 with the classical
Jarník–Besicovitch theorem, one might ask if the lower dimension can be
replaced by the Hausdorff dimension, or indeed if a lower bound of the form
$(\dim_{\textup{L}}F+1)/t$ or $c\dim_{\textup{H}}F/t$ is possible for some
uniform constant $c$. The following proposition shows that these bounds are
not possible in general and demonstrates that $F_{t}$ ($t>1$) can be very
small, even countable, if the abstract rationals are “far away” from the large
parts of $F$.

######  Proposition 2.4.

There exists a compact set $F\subseteq[0,1]$ with
$\dim_{\textup{H}}F=\dim_{\textup{P}}F>0$ and such that, for some choice of
abstract rationals, $F_{t}$ is countable for all $t>1$ and, for another choice
of abstract rationals,

$\dim_{\textup{H}}F_{t}>0$

for all $t>1$.

  * Proof.

Let $F_{0}$ be the middle third Cantor set (see [F14, Example 4.5]) and $F$ be
$F_{0}\cup C$ where $C$ is the countable set consisting of a single point in
the centre of each complementary interval of $F_{0}$. Clearly
$\dim_{\textup{H}}F=\dim_{\textup{P}}F=s=\log_{3}2>0$. We choose the first set
of abstract rationals such that $\cup_{q}P(Q)=C$. More precisely, for
$q\geqslant 1$, let $C_{k}\subseteq C$ be the set of centres of complementary
intervals of length $3^{-k}$ where $k$ is chosen such that
$3^{-(k-1)}/2\leqslant 1/q<3^{-(k-2)}/2$. Then choose $P(q)$ to be a maximal
$(1/q)$-separated subset of $C$ which contains $C_{k}$. Note that $P(q)$ is a
maximal $(1/q)$-separated subset of $F$ by construction. Fix $t>1$,
$q\in\mathbb{N}$ and $p\in P(q)$ and let $k$ be as above. Then

$B(p,q^{-t})\subseteq B(p,3^{-t(k-2)}/2^{t})=\\{p\\}$

for sufficiently large $q$. More precisely, once $q$ is large enough so that
$k$ is large enough to guarantee

$3^{-t(k-2)}/2^{t}<3^{-k}/2.$

It follows that $F_{t}=C$ which is countable and, in particular,

$\dim_{\textup{H}}F_{t}=\dim_{\textup{P}}F_{t}=0.$

For the second choice of abstract rationals, choose $P(q)$ such that it
contains a maximal $(1/q)$-separated subset of $F_{0}$, which we denote by
$P_{0}(q)\subseteq F_{0}$. Let $F_{0,t}=(F_{0})_{t}\subseteq F_{0}$ be the set
of points in $F_{0}$ which are i.o. $t$-approximable by abstract rationals
$P_{0}(q)$. Then, since $F_{0}$ is $s$-Ahlfors–David regular, it follows from
Theorem 2.7, which we prove later, that

$\dim_{\textup{H}}F_{t}\geqslant\dim_{\textup{H}}F_{0,t}\geqslant s/t>0$

as required. ∎

The next result is an upgrade of the previous example to show that the lower
bound $\dim_{\textup{L}}F/t$ is sharp in a stronger sense.

######  Proposition 2.5.

For all $0\leqslant L\leqslant H\leqslant 1$, there exists a compact set
$F\subseteq[0,1]$ with

$\dim_{\textup{L}}F=L$

and

$\dim_{\textup{H}}F=\dim_{\textup{P}}F=H$

such that for some choice of abstract rationals,

$\dim_{\textup{H}}F_{t}=L/t=\frac{\dim_{\textup{L}}F}{t}$

for all $t>1$.

  * Proof.

This proposition is proved in the same way as Proposition 2.4 but with the
middle third cantor set replaced by a central (self-similar) cantor set of
dimension $H$ and with the single points placed in the complementary intervals
replaced by scaled copies of a fixed set $E$ with
$\dim_{\textup{L}}E=\dim_{\textup{H}}E=L$. The set $E$ could be another
central cantor set, for example. The scaling is done such that the diameter of
a scaled copy is much smaller than the length of the complementary interval it
is placed in. The abstract rationals are chosen to be contained in the
countably many copies of $E$ and within each copy $E^{\prime}$ of $E$ must be
chosen such that

(3) $\dim_{\textup{H}}E^{\prime}_{t}=\frac{\dim_{\textup{H}}E^{\prime}}{t}.$

The details work the same, but the challenge is in finding a way to choose the
set $E$ and abstract rationals to satisfy (3). We choose $E$ to be an
$L$-Ahlfors–David regular set with the abstract rationals chosen as in
Proposition 3.4, which we prove later. ∎

There are many other ways in which one could investigate sharpness of Theorem
2.1. For example, our results above are pointwise in the sense that for a
given $t$ a set $F$ is constructed. It may be interesting to consider if one
can construct sets witnessing sharpness for all $t\geqslant 1$ simultaneously.
Another question of sharpness which we have not answered is if the _Hausdorff_
dimension of $F_{t}$ can be estimated from below in general.

######  Question 2.6.

Is it true that for all $t\geqslant 1$

$\dim_{\textup{H}}F_{t}\geqslant\frac{\dim_{\textup{L}}F}{t}?$

###  2.3. The Ahlfors–David regular case

It turns out we can say more when $F$ is an Ahlfors–David regular metric
space. Recall that a compact set $F$ is $s$-Ahlfors–David regular for some
$s\geqslant 0$ if there exists a constant $C>0$ such that

$r^{s}/C\leqslant\mathcal{H}^{s}(B(x,r))\leqslant Cr^{s}$

for all $0<r<1$ and all $x\in F$. In this case it is not hard to see that

$\dim_{\textup{L}}F=\dim_{\textup{H}}F=\dim_{\textup{P}}F=\overline{\dim}_{\textup{B}}F=s$

and that $F$ is doubling, see [F20, Theorem 6.4.1].

######  Theorem 2.7.

Suppose $F$ is a compact $s$-Ahlfors–David regular. For all $t\geqslant 1$

$\frac{s}{t}\leqslant\dim_{\textup{H}}F_{t}\leqslant\frac{s+1}{t}.$

  * Proof.

Given the general bounds, all that remains to establish is

$\dim_{\textup{H}}F_{t}\geqslant\frac{s}{t}$

which we prove now. The strategy is similar to the general lower bound, except
that we need to control the measure at all scales, not just a sequence of
scales. Let $E$ and $\mu$ be as in the proof of Theorem 2.1. Given
$r\in(0,q_{1}^{-t})$, let $k$ be chosen uniquely such that

$q_{k+1}^{-1}\leqslant r<q_{k}^{-1}.$

Since $t\geqslant 1$ is fixed and the $q_{k}$ are rapidly increasing, we may
assume that either

(4) $q_{k+1}^{-1}\leqslant r<q_{k}^{-t}\leqslant q_{k}^{-1}$

or

(5) $q_{k+1}^{-1}<q_{k}^{-t}\leqslant r<q_{k}^{-1}.$

Consider the first case (4). There is a uniform constant $C_{3}$ such that an
arbitrary ball $B(x,r)$ may intersect $C_{3}(r/q_{k+1}^{-1})^{s}$ many
$(k+1)$-level cylinders. This uses the fact that $P(q_{k+1})$ is
$(1/q_{k+1})$-separated and the fact that $F$ is $s$-Ahfors-David regular.
Moreover, each $(k+1)$-level cylinder carries mass at most

(6) $q_{1}^{ts}(q_{2}\cdots q_{k})^{s(t-1)}(q_{k+1}^{-1})^{s}$

by following the argument above (Proof.). Therefore

$\displaystyle\mu(B(x,r))$ $\displaystyle\leqslant$ $\displaystyle
C_{3}(r/q_{k+1}^{-1})^{s}q_{1}^{ts}(q_{2}\cdots
q_{k})^{s(t-1)}(q_{k+1}^{-1})^{s}\qquad\text{by \eqref{dunkel}}$
$\displaystyle=$ $\displaystyle C_{3}q_{1}^{ts}(q_{2}\cdots
q_{k})^{s(t-1)}\,r^{s}$ $\displaystyle\leqslant$ $\displaystyle
C_{3}\left(\log(q_{k})\right)^{ts}q_{k}^{s(t-1)}\,r^{s}\qquad\text{by
\eqref{rapid}}$ $\displaystyle\leqslant$ $\displaystyle C_{3}|\log
r|^{ts}\left(r^{-1/t}\right)^{s(t-1)}\,r^{s}\qquad\text{by \eqref{case1}}$
$\displaystyle=$ $\displaystyle C_{3}|\log r|^{ts}r^{s/t}.$

Now consider the second case (5). Since $F$ is Ahlfors–David regular it is
doubling and, therefore, there exists a uniform constant $C_{4}$ such that an
arbitrary ball $B(x,r)$ intersects at most $C_{4}$ many $k$-level cylinders.
This uses the fact that the centres of the $k$-level cylinders are
$(1/q_{k})$-separated. Therefore, using (6) to estimate the mass of a
$k$-level cylinder,

$\displaystyle\mu(B(x,r))$ $\displaystyle\leqslant$ $\displaystyle
C_{4}q_{1}^{ts}(q_{2}\cdots q_{k-1})^{s(t-1)}(q_{k}^{-1})^{s}$
$\displaystyle\leqslant$ $\displaystyle
C_{4}\left(\log(q_{k})\right)^{ts}(q_{k}^{-1})^{s}\qquad\text{by
\eqref{rapid}}$ $\displaystyle\leqslant$ $\displaystyle C_{4}|\log
r|^{ts}r^{s/t}\qquad\text{by \eqref{case2}.}$

Combining the two cases, we have proved that for an arbitrary $x\in E$ and
$r\in(0,q_{1}^{-t})$

$\mu(B(x,r))\leqslant\max\\{C_{3},C_{4}\\}|\log r|^{ts}r^{s/t}$

and the result follows by the mass distribution principle, see [F14, Chapter
4]. ∎

##  3\. Dirichlet spectrum of Ahlfors–David regular spaces

Notice that in the classical setting of Diophantine approximation on $[0,1]$
with $P$ defined by

$P(q)=\left\\{0,1/q,2/q,\dots,1\right\\},\qquad(q\geqslant 1),$

we have $s=1$ and the _upper_ bound in Theorem 2.7 gives the correct dimension
of $F_{t}$, namely, $\dim_{\textup{H}}F_{t}=2/t$, the dimension given in the
Jarník–Besicovitch theorem. Therefore probing sharpness of the lower bound
seems like the most pressing concern.

Notice also that in the classical setting, we have

$\displaystyle
2=\sup\left\\{d:F_{d}=[0,1]\right\\}=\inf\left\\{d:\operatorname{Leb}(F_{d})=0\right\\}$
$\displaystyle=\sup\left\\{d:F_{d}\textrm{ contains a
ball}\right\\}=\sup\left\\{d:F_{d}\textrm{ essentially contains a
ball}\right\\}.$

As a generalisation of this, define the following: for a given
$s$-Ahlfors–David regular set $F$ and choice $P$ of abstract rationals in $F$,
define the _Dirichlet exponent_

$\displaystyle d(P)$ $\displaystyle=\sup\left\\{d:F_{d}\textrm{ contains a
ball up to $\mathcal{H}^{s}$-measure $0$}\right\\}$
$\displaystyle=\sup\left\\{d:(\exists\textrm{ball }B\subset
F)\quad\mathcal{H}^{s}(B\setminus F_{d})=0\right\\}$

and the _Dirichlet spectrum_

$\sigma(F)=\left\\{d(P):P\textrm{ is a choice of abstract rationals in
}F\right\\}.$

We can then ask the following.

######  Question 3.1.

For a given $s$-Ahlfors–David-regular $F$, what is the nature of the set
$\sigma(F)$, and to what extent does the formula

$\dim_{\textup{H}}F_{t}=\frac{d(P)s}{t}\qquad(t\geqslant d(P))$

hold for a given choice $P$ of abstract rationals in $F$?

In exploring this question, we will find, in particular, that the dimension
bounds in Theorem 2.7 are sharp. First, let us report what Theorem 2.7 and the
Mass Transference Principle say immediately about Question 3.1.

######  Proposition 3.1.

If $F$ is a locally compact, $s$-Ahlfors–David-regular metric space, then
$\sigma(F)\subset[1,1+1/s]$, and for any choice $P$ of abstract rationals in
$F$ we have

$\dim_{\textup{H}}F_{t}\geqslant\frac{d(P)s}{t}\qquad(t\geqslant d(P)).$

  * Proof.

Clearly, since for any $P$ one has $F=F_{1}$, we have $d(P)\geqslant 1$. On
the other hand, suppose $\mathcal{H}^{s}(F_{d})>0$. Then
$\dim_{\textup{H}}F_{d}\geqslant s$, which, by Theorem 2.7, implies that
$s\leqslant(s+1)/d$, hence $d\leqslant 1+1/s$. Taking supremum over all such
$d$ shows that $d(P)\leqslant 1+1/s$, so we have shown $d(P)\in[1,1+1/s]$ for
all $P$.

Now suppose $d<d(P)\leqslant t$. It follows that $F_{d}$ essentially contains
a closed ball $F^{\prime}=B(x^{\prime},r)\subset F$, and we may suppose that
$\mathcal{H}^{s}(S(x^{\prime},r))=0$, where $S(x^{\prime},r)$ denotes the
sphere of radius $r$ centered at $x^{\prime}$. (After all, at most countably
many members of the uncountable disjoint union
$F^{\prime}=\bigcup_{0\leqslant\rho\leqslant r}S(x^{\prime},\rho)$ may have
positive measure, so if we must, we may shrink $r$ until $S(x^{\prime},r)$ has
zero measure.) The ball $F^{\prime}$ is itself an $s$-Ahlfors–David-regular
metric space. Define a hierarchy of points $P^{\prime}=P\mid_{F^{\prime}}$ in
$F^{\prime}$ by

$P^{\prime}(q)=\begin{cases}P(q)\cap F^{\prime}&\textrm{if }P(q)\cap
F^{\prime}\neq\emptyset\\\ x^{\prime}&\textrm{if }P(q)\cap
F^{\prime}=\emptyset.\end{cases}$

Note that as soon as $q$ is large enough we have $P^{\prime}(q)=P(q)\cap
F^{\prime}$, hence $P^{\prime}(q)$ is $(1/q)$-separated but it may not be
maximal. That is, $P^{\prime}$ may not constitute abstract rationals in
$F^{\prime}$ as we have defined the term. Still, let us use $F_{t}^{\prime}$
to denote all the points in $F^{\prime}$ that are $t$-approximable by
$P^{\prime}$. Notice then that

(7) $(F_{d}\cap F^{\prime})\setminus F_{d}^{\prime}\subset S(x^{\prime},r),$

for, if $x\in F_{d}$ and $d(x^{\prime},x)<r$, then for all $q$ large enough,
every $p\in P(q)$ with $d(x,p)\leqslant q^{-d}$ must lie in $F^{\prime}$. All
such $p$ are in $P^{\prime}(q)$, so we have $x\in F_{d}^{\prime}$. Importantly
for us, (7) implies that $\mathcal{H}^{s}(F^{\prime}\setminus
F_{d}^{\prime})=0$.

For any $t\geqslant d(P)$, we have $F_{t}^{\prime}\subset F_{t}$. Also, since
$\mathcal{H}^{s}(F^{\prime}\setminus F_{d}^{\prime})=0$, we have that for any
ball $B\subset F^{\prime}$,

$\mathcal{H}^{s}(F_{d}^{\prime}\cap B)=\mathcal{H}^{s}(B).$

So the mass transference principle [BV06, Theorem 3] implies that for any ball
$B\subset F^{\prime}$ we have

$\mathcal{H}^{ds/t}(F_{t}^{\prime}\cap B)=\mathcal{H}^{ds/t}(B)=\infty.$

Therefore $\dim_{\textup{H}}F_{t}^{\prime}\geqslant ds/t$, hence
$\dim_{\textup{H}}F_{t}\geqslant ds/t$. Since $d<d(P)$ was arbitrary, the
result follows. ∎

######  Remark.

We could of course ask Question 3.1 when $F$ is not Ahlfors–David regular.
There, Theorem 2.1 would guarantee that

$1\leqslant d(P)\leqslant\frac{\dim_{\textup{P}}F+1}{\dim_{H}F}.$

However, clearly no such dimension formula as the one in Question 3.1 holds in
general, even with $s$ replaced by $\dim_{\textup{H}}F.$

We can now refine Question 3.1 by asking what kind of subset
$\sigma(F)\subset[1,1+1/s]$ is. The next results exhibit cases where
$\sigma(F)=[1,1+1/s]$. We will also show that for certain spaces $F$ and any
$d\in\sigma(F)$, there exist abstract rationals witnessing $d(P)=d$ such that
$\dim_{\textup{H}}F_{t}=ds/t$ for all $t\geqslant d$. That is, the full
spectrum $\sigma(F)$ can be realized by abstract rationals satisfying a
general Jarnik–Besicovitch theorem.

In particular, we show that the bounds from Theorem 2.7 are indeed sharp and,
moreover, the dimensions of $F_{t}$ can depend on the choice of abstract
rationals, even in the Ahlfors–David regular setting.

######  Proposition 3.2.

For $F=[0,1]$ we have $\sigma(F)=[1,2]$ and for any $d\in\sigma(F)$ there is a
choice $P$ of abstract rationals with $d(P)=d$ such that

$\dim_{\textup{H}}F_{t}=\frac{d(P)}{t}\qquad(t\geqslant d(P)).$

  * Proof.

The case $d(P)=2$ is achieved by the classical setting of the
Jarník–Besicovitch theorem so it remains to consider $d\in[1,2)$. We first
prove the case $d=1$, which provides a roadmap to the other cases.

For each $q\in[2^{k},2^{k+1})$ let

$P(q)=\left\\{\frac{a}{2^{k}}:a=0,\dots,2^{k}\right\\}$

be the rationals with denominator $2^{k}$. Note that $P(q)$ is a maximal
$(1/q)$-separated set in $F$. In this setting we have $d(P)=1$, which in
particular shows $1\in\sigma(F)$. This will follow from Proposition 3.1
together with the upper bound for the dimension of $F_{t}$ which we prove now.
The lower bound

$\dim_{\textup{H}}F_{t}\geqslant\frac{1}{t}\qquad(t\geqslant 1),$

already follows from Theorem 2.7. Note that

$F_{t}\subset\bigcup_{k\geqslant
1}\bigcup_{a=0}^{2^{k}}\left[\frac{a}{2^{k}}-\frac{1}{2^{tk}},\frac{a}{2^{k}}+\frac{1}{2^{tk}}\right].$

For all $\tau>1/t$ we have that

$\sum_{k\geqslant
1}\sum_{a=0}^{2^{k}}\left\lparen\frac{1}{2^{tk-1}}\right\rparen^{\tau}\ll\sum_{k\geqslant
1}2^{(1-\tau t)k},$

which converges. (The notation $\ll$ means that $\leqslant$ holds up to a
positive absolute constant factor.) This implies
$\mathcal{H}^{\tau}(F_{t})<\infty$ and $\dim_{\textup{H}}F_{t}\leqslant\tau$.
Therefore $\dim_{\textup{H}}F_{t}\leqslant 1/t$.

Now suppose $d\in(1,2)$ and let $\alpha>1$ be such that $d=1+1/\alpha$. For
$q\in\mathbb{N}$ with $k^{\alpha}\leqslant q<(k+1)^{\alpha}$, let

$P(q)=\left\\{\frac{a}{\lfloor k^{\alpha}\rfloor}:a=0,\dots,\lfloor
k^{\alpha}\rfloor\right\\}.$

We claim that $d(P)=d$. In fact, this is a consequence of the recently proved
Duffin–Schaeffer conjecture [KM], which states that if
$\psi(n):\mathbb{N}\to\mathbb{R}$ is such that

$\sum_{q}\psi(q)\varphi(q)=\infty,$

then for almost every $x\in[0,1]$ there are infinitely many $p/q\in\mathbb{Q}$
such that

$\left\lvert x-\frac{p}{q}\right\rvert<\psi(q).$

Here and throughout $\varphi$ is the is the Euler totient function. Consider
the function

$\psi(q)=\begin{cases}q^{-d}&\textrm{if }q=\left\lfloor
k^{\alpha}\right\rfloor,k\in\mathbb{N}\\\ 0&\textrm{otherwise.}\end{cases}$

Then

$\displaystyle\sum_{q}\psi(q)\varphi(q)$
$\displaystyle=\sum_{k}\frac{1}{\lfloor k^{\alpha}\rfloor^{d}}\varphi(\lfloor
k^{\alpha}\rfloor)$ $\displaystyle\geqslant\sum_{k}\frac{\varphi(\lfloor
k^{\alpha}\rfloor)}{k^{\alpha+1}}$
$\displaystyle\geqslant\sum_{k}\frac{\varphi(\lfloor
k^{\alpha}\rfloor)}{k^{\alpha+1}}$ $\displaystyle\gg\sum_{k}\frac{1}{k\log\log
k^{\alpha}}=\infty,$

this last estimate coming from the growth estimate $\varphi(n)\gg n/\log\log
n$ [HW]. We may then conclude that for almost every $x\in[0,1]$ there are
infinitely many $q$ such that $x$ is within $q^{-d}$ of some $p\in P(q)$. That
is, $F_{d}$ has full measure, hence $d(P)\geqslant d$. On the other hand, if
we had defined $\psi$ by $q^{-d^{\prime}}$ for some $d^{\prime}>d$, then the
above sum would have converged, and the Borel–Cantelli lemma would imply that
$F_{d^{\prime}}$ had measure $0$. This shows that $d(P)=d$, as claimed.

Proposition 3.1 now gives

$\dim_{\textup{H}}F_{t}\geqslant\frac{d(P)}{t}$

for every $t\geqslant d$. The corresponding upper bound comes from a cover of
$F_{t}$ by intervals as in the previous case. Here the relevant series is

$\sum_{k\geqslant 1}\sum_{a=0}^{\lfloor k^{\alpha}\rfloor}\left\lparen
k^{-\alpha t}\right\rparen^{\tau}\ll\sum_{k\geqslant 1}k^{\alpha(1-t\tau)},$

which converges as long as

$\tau>\frac{1+1/\alpha}{t}=\frac{d}{t},$

which leads to the upper bound and establishes

(8) $\dim_{\textup{H}}F_{t}=\frac{1+1/\alpha}{t}=\frac{d(P)}{t}.$

Since $d\in(1,2)$ was arbitrary, we are done. ∎

We investigate Dirichlet exponents for examples of Ahlfors–David regular $F$
with dimension less than 1.

######  Proposition 3.3.

Let $F$ be the middle third Cantor set, whose dimension is $s=\log_{3}2$. Then
for any $d\in\left[1,1+1/s\right]$ there exists a choice of abstract rationals
such that $d(F)=d$ and

$\dim_{\textup{H}}F_{t}=\frac{d}{t}s\quad(t\geqslant d).$

  * Proof.

(i) _The case $d=1$._ First, for $3^{n}\leqslant q<3^{n+1}$ let $P(q)$ be the
endpoints of the basic intervals in the $n$th construction set $E_{n}$.
Theorem 2.7 tells us that for any $t\geqslant 1$, $\dim_{H}F_{t}\geqslant
s/t$. For the upper bound, note that $F_{t}$ is covered by

$F_{t}\subset\bigcup_{n\geqslant m}\bigcup_{p\in
P(3^{n})}\left[p-3^{-nt},p+3^{-nt}\right]$

for any $m>0$. For $\tau\geqslant 0$, the sum of diameters is

$\sum_{n\geqslant m}2^{n+2}3^{-nt\tau},$

which converges as long as $2<3^{t\tau}$, that is, if $\tau>s/t$. This shows
the other inequality in dimension, hence $\dim_{H}F_{t}=s/t$.

Notice that in this example we have that for every $x\in F$ there are
infinitely many $q$ for which there is $p\in P(q)$ satisfying $\lvert
x-p\rvert\leqslant 1/(3q)$. The previous paragraph shows that as soon as we
increase the exponent beyond $1$, the dimension drops. Therefore, the $d(F)=1$
with this choice of $P$.

(ii) _The case $d=1+1/s$. _ For $n\geqslant 0$, let

$k(n)=k=\left\lfloor 1+n\left\lparen 1+\frac{1}{s}\right\rparen\right\rfloor,$

that is, $k(n)$ is the greatest integer with the property that each basic
interval of $E_{n}$ contains at most $3^{n+1}-3^{n}$ basic intervals of
$E_{k}$. Denote by

$E_{n}^{L}=\bigcup_{\begin{subarray}{c}I\subset E_{n}\\\ \textrm{basic
interval}\end{subarray}}\left\\{\min
I\right\\}=\\{\ell_{1},\dots,\ell_{2^{n}}\\}$

the set of left endpoints $\ell_{i}$ of the basic intervals of $E_{n}$. For
$3^{n}\leqslant q<3^{n+1}$ put

$m=\left\lparen q-3^{n}\quad(\bmod 2^{k-n})\right\rparen+1$

and define

$P(q)=\left\lparen E_{n}^{L}+\ell_{m}\right\rparen\cup\left\\{\textrm{enough
endpoints of $E_{n}$ to get maximality}\right\\}.$

In particular, $P(3^{n})$ consists exactly of the endpoints of the basic
intervals of $E_{n}$, as in the previous example. (See Figure 1 for an
illustration of $P(1)$ throught $P(9)$.) Importantly, we have, for
$3^{n}\leqslant q<3^{n+1}$

$2^{n-1}=\frac{\\#P(3^{n})}{2}\leqslant\\#P(q)\leqslant\\#P(3^{n})=2^{n}.$

The “Dirichlet theorem” here is: For every $x\in F$ and $n\in\mathbb{N}$,
there exists $q\leqslant 3^{n+1}$ and $p\in P(q)$ such that

$\left\lvert x-p\right\rvert\leqslant 3^{-k}\leqslant\frac{1}{q^{k/(n+1)}}.$

Since

$\frac{k(n)}{n+1}\sim 1+\frac{1}{s},$

the Dirichlet exponent here is $d(F)=1+1/s$.

By Proposition 3.1 we have $\dim_{H}F_{t}\geqslant d(P)s/t=\frac{s+1}{t}$ for
all $t\geqslant d(P)$. For the upper bound, we cover

$F_{t}\subset\bigcup_{q\geqslant 3^{m}}\bigcup_{p\in
P(q)}B(p,q^{-t})\subset\bigcup_{q\geqslant 3^{m}}\bigcup_{p\in
P(q)}B(p,3^{-nt})$

The corresponding sum of diameters, with $\tau>0$, is

$\sum_{q\geqslant 3^{m}}\sum_{p\in P(q)}3^{-nt\tau}=\sum_{n\geqslant
m}2^{k}3^{-nt\tau}\leqslant\sum_{n\geqslant m}2^{n(1+1/s)}3^{-nt\tau},$

which converges if $\tau>(s+1)/t$. This leads to the desired upper bound, so
we have

$\dim_{H}F_{t}=\frac{s+1}{t}=\frac{d(F)}{t}s.$

$P(1)$$P(2)$$P(3)$$P(4)$$P(5)$$P(6)$$P(7)$$P(8)$$P(9)$$\vdots$$\vdots$$\vdots$$\vdots$
Figure 1. The red dots are the elements of $\left\lparen
E_{n}^{L}+\ell_{m}\right\rparen$ and the blue dots are the endpoints of
$E_{n}$ we have included to achieve maximality. United they are $P(q)$.

(iii) _The intermediate case._ Let $d\in(1,1+1/s)$. We now modify the previous
construction by using instead

$k(n)=k=\left\lfloor nd\right\rfloor$

and keeping the rest of the definitions intact. The same Dirichlet theorem
from the previous construction continues to hold, but now we have

$\frac{k(n)}{n+1}\sim d,$

so that $d(F)\geqslant d$.

For the other bound, let $d^{\prime}>d$, and note that $F_{d^{\prime}}$ is
covered by

$F_{d^{\prime}}\subset\bigcup_{q\geqslant q_{0}}\bigcup_{p\in
P(q)}B\left\lparen p,q^{-d^{\prime}}\right\rparen$

for any $q_{0}$, and therefore the measure of $F_{d^{\prime}}$ is bounded by

$\displaystyle\sum_{q\geqslant q_{0}}\sum_{p\in P(q)}q^{-d^{\prime}s}$
$\displaystyle\leqslant\sum_{n\geqslant n_{0}}\sum_{3^{n}\leqslant
q<3^{n+1}}\sum_{p\in P(q)}q^{-d^{\prime}s}$
$\displaystyle\leqslant\sum_{n\geqslant n_{0}}\sum_{3^{n}\leqslant
q<3^{n+1}}2^{n}3^{-nd^{\prime}s}$ $\displaystyle\leqslant\sum_{n\geqslant
n_{0}}2^{k-n}2^{n}3^{-nd^{\prime}s}$ $\displaystyle=\sum_{n\geqslant
n_{0}}3^{(k-nd^{\prime})s}.$

This sum converges because $k/n\sim d$ and $d^{\prime}>d$, so by sending
$n_{0}\to\infty$ we see that $F_{d^{\prime}}$ has zero measure. Therefore
$d(F)=d$ as wanted.

Let $t\geqslant d$. Then, by Proposition 3.1
$\dim_{\textup{H}}F_{t}\geqslant\frac{ds}{t}$. For the upper upper bound we
cover

$F_{t}\subset\bigcup_{n\geqslant 1}\underbrace{\bigcup_{3^{n}\leqslant
q\leqslant 3^{n+1}}}_{\textrm{redundant}}\bigcup_{p\in P(q)}B(p,q^{-t})$

and observe that the indicated union has a redundancy in it owing to the fact
that as $q$ ranges through $(3^{n},3^{n+1})\cap\mathbb{Z}$, $P(q)$ cycles with
period $2^{k-n}$. That is, we may more efficiently cover by

$F_{t}\subset\bigcup_{n\geqslant
1}\bigcup_{q=3^{n}}^{3^{n}+2^{k-n}}\bigcup_{p\in P(q)}B(p,q^{-t})$

The diameter sum is

$\displaystyle\sum_{n\geqslant 1}\sum_{q=3^{n}}^{3^{n}+2^{k-n}}\sum_{p\in
P(q)}\text{diam}\left\lparen B(p,q^{-t})\right\rparen^{\tau}$
$\displaystyle\ll\sum_{n\geqslant 1}\sum_{q=3^{n}}^{3^{n}+2^{k-n}}\sum_{p\in
P(q)}q^{-t\tau}$ $\displaystyle=\sum_{n\geqslant
1}\sum_{q=3^{n}}^{3^{n}+2^{k-n}}\\#P(q)q^{-t\tau}$
$\displaystyle\leqslant\sum_{n\geqslant 1}2^{k-n}\\#P(3^{n})3^{-nt\tau}$
$\displaystyle\leqslant\sum_{n\geqslant 1}2^{k}3^{-nt\tau}$
$\displaystyle\leqslant\sum_{n\geqslant 1}3^{ks-nt\tau}.$

This sum converges if $\tau>ds/t$, therefore $\dim_{\textup{H}}F_{t}\leqslant
ds/t$ and we are done. ∎

The above result holds for more general Ahlfors–David regular sets than just
the middle third Cantor set. Since it is especially useful to have examples
with different dimensions having minimal Dirichlet exponent, see Proof of
Proposition 2.5, we record the following partial extension.

######  Proposition 3.4.

Let $F$ be the central Cantor set having dimension $s=-\log 2/\log\lambda$,
where $\lambda\in(0,1/2)$. Then there is a choice $P$ of abstract rationals
such that $\dim_{\textup{H}}F_{t}=s/t$ for all $t\geqslant 1$.

  * Proof.

The proof of this statement is essentially the same as the proof of
Proposition 3.3 (i). Using the analogous notation, for $\lambda^{-n}\leqslant
q<\lambda^{-n-1}$ let

$P(q)=E_{n}^{L}\cup\left\\{\textrm{enough endpoints of $E_{n}$ to get
maximality}\right\\}.$

Notice that $2^{n-1}\leqslant P(q)\leqslant 2^{n}$. Notice that
$P(\lambda^{-n})=E_{n}^{L}$ and every point of $F$ is in the
$\lambda^{n}$-neigbourhood of every $P(\lambda^{-n})$, so that $d(F)\geq 1$.

What is more, for any $m$,

$F_{t}\subset\bigcup_{n\geq m}\bigcup_{p\in
P(\lambda^{-n})}[p-\lambda^{-nt},p+\lambda^{-nt}].$

This implies that

$\mathcal{H}^{\tau}(F_{t})\leq\sum_{m\geq n}2^{n}\lambda^{-nt\tau},$

which converges if $\tau>s/t$. This implies two things: for $t>1$, we
immediately obtain $\dim_{\textup{H}}F_{t}<\dim_{\textup{H}}F$ so that
$d(F)=1$. Further, we also see that $\dim_{\textup{H}}F_{t}\leq s/t$. Since
Theorem 2.7 tells us that $\dim_{\textup{H}}F_{t}\geqslant s/t$, we have
finished the proof of the theorem. ∎

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# Coherent Transport in Y-Junction Graphene Waveguide

Vahid Mosallanejad<EMAIL_ADDRESS>Kuei-Lin Chiu  Guo-ping Guo
<EMAIL_ADDRESS>CAS Key Laboratory of Quantum Information, and Synergetic
Innovation Center of Quantum Information and Quantum Physics, University of
Science and Technology of China,Chinese Academy of Sciences, Hefei
230026,China

###### Abstract

We performed a series of theoretical transport studies on Y-branch electron
waveguides which are embedded in mid-size armchair graphene nanoribbons
(AGNRs). Non-equilibrium Green’s function (NEGF) with different approximations
of tight-binding (TB) Hamiltonian has been employed. Using the first nearest
hopping approximation, we observed very pronounced conductance quantization,
the structure of which depends on geometrical design and shows a spacing of
$4e^{2}/h$, indicating the existence of valley degree of freedom. Moreover, by
incorporating the third nearest approximation, we observed seminal plateaus
deviated from multiples of $4e^{2}/h$ conductance, suggesting the lift of
valley degeneracy. Finally, Quasi-one dimensional band structure calculations
have been performed to study the availability of energy channels and the role
of the major geometrical parameters on the transport.

††preprint: APS/123-QED

##  I  Introduction

In the past decade, many theoretical studies have been proposed to realize the
coherent transports in ideal straight graphene nanoribbons (GNRs)
peres2006conductance ; gunlycke2007semiconducting ; dubois2009electronic .
However, the quantization of conductance, which is the hallmark of coherent
transport in 1D systems, is usually missing in $O_{2}$ plasma etched GNRs due
to the edge disorders that produce strong scattering lin2008electrical ;
lian2010quantum ; tombros2011quantized ; wurm2009interfaces ;
mucciolo2009conductance ; lherbier2008transport ; orlof2013effect ;
baldwin2016effect . Perhaps this major problem has hindered graphene’s
application in sophisticated nanoelectronics (such as spin qubit devices)
despite its excellent conductivity recher2010quantum ; chiu2017single ;
deng2015charge . An alternative approach to replicate a GNR is to introduce a
well-like electrostatic confinement in graphene, also known as graphene
waveguide (GWG), in which the crystal structure between side-barriers and
quantum well remains intact tudorovskiy2007spatially. As a result of the
electrostatic confinement, there are discrete bound states spatially extended
along the graphene waveguide. Quantized conductance in 1D systems had long
been realized in modulation-doped GaAs/AlGaAs heterostructures where a pair of
negatively biased split gates deplete the two-dimensional electron gas (2DEG)
underneath and define a 1D channel yacoby1996nonuniversal . The situation for
graphene-based waveguide is expected to be different from the aforementioned
2DEG case, since the existence of transparent Klein tunneling could play an
essential role and limit the ability of GWG for efficient guiding
katsnelson2006chiral ; allain2011klein . Interestingly, some theoretical
approaches using Dirac equation have in fact suggested that carriers in an
infinite GWG is behaving in many respects like the optical waves in an
optical-fiber (that is, they display optics-like features)
beenakker2009quantum ; hartmann2014quasi ; dragoman2010polarization . However,
it is worth noting that the theory of Klein tunneling on graphene is based on
first-nearest approximation (Dirac Hamiltonian) and the assumption of the
plane wave solution. A precise but relatively expensive numerical method
incorporating few more nearest-neighbors hopping terms, called the Non-
equilibrium Green’s function (NEGF), can be employed to have a better
understanding about the feasibility of the perfect transmission in GWG
datta2005quantum . Such a numerical simulation can also provide more details
on the conditions that lead to a coherent or imperfect transmission. Current
experimental evidences are not, however, in favor of the total reflection of
carriers from borders of a GWG as Snell’s law would hold in the optical-fiber.
In particular, the reported guiding efficiencies do not exceed 80% and differ
among different 1D energy modes williams2011gate ; rickhaus2015guiding . In
our previous calculations, we have shown that a single GWG (S-GWG) can possess
the characteristics of a coherent transport and the desirable insensitivity to
the bending degree mosallanejad2018perfectly. A further interesting question
for us is whether a coherent splitting of current can be realized in GWGs. In
this letter, we provide theoretical evidences to show that it is possible to
coherently split the incoming current into two paths in a Y-junction graphene
waveguide (YJ-GWG). Our results show the prospect of using YJ-GWGs as a
splitter in all graphene-based electronic devices that require robust phase
coherence, and therefore can be potentially used as an interconnect in
spintronic devices. A YJ-GWG can be made by applying a positive potential to
an external Y-branch metal gate which is isolated from graphene flake and
consequently it induces a Y-shape potential well in graphene. Mathematically,
a reasonable estimation of 2D on-site potential (a cut-plan of 3D
electrostatic potential) suitable for modeling of YJ-GWG could be produce by
making use of two bended S-GWG with an overlapping area as it explained in
appendix A. The influence of geometrical design on the coherent transport is
discussed in detail in the main text. Furthermore, Quasi-one dimensional
bandstructure calculations have been performed to study the energy channels
that could contribute to the transport.

##  II Methods

###  II.1 Device structure

Figure 1:  (a) Schematic illustration of YJ-GWG where a Y-junction metal gate
is embedded in an insulator and a mid-size GNR lies above. Enlarged view shows
hopping parameters. The crystal structure of graphene is scaled for
illustrative purposes and do not represent the realistic crystal size.(b) (c)
Realistic on-site potential of the proposed structure in Fig. 1(a). Source and
drains are shown by vertical (red) lines and their widths are denoted by WS
and W${}_{D_{1,2}}$.

The schematic of our proposed structure is shown in Fig. 1(a), which consists
of a finite mid-size AGNR with an integrated Y-junction gate underneath it.
Hexagonal boron nitride (h-BN), an atomically thin insulating material, is
used simultaneously as a gate dielectric and a secondary substrate to improve
the mobility of graphene zomer2011transfer ; zhang2017electrotunable . A YJ-
GWG can be defined by applying a positive potential to the underlying gate and
consequently induces a Y-shape potential well in the AGNR. Single potential
well on the left side thus splits into two wells, one goes upward and the
other goes downward beyond the intersection, i.e., the system has in total
three terminals. The source and drain terminals are located exactly above the
respective branches of the Y-junction, which is denoted by S, D1 and D2,
respectively. In general, a self-consistent solution of Schrödinger equation
(in the form of Green’s function formalism) coupling to a Poissons equation
should be used to simulate the electrostatics of the device and the
transmission through terminals, simultaneously akhavan2012phonon ;
banadaki2015investigation . However, this model can not be realized here due
to the numerical complexity arising from the large number of atoms in our
system. To compensate the lack of full self-consistent solution, a
representative potential well with a cross section as illustrated in Fig. 1(b)
has been used to define YJ-GWG in our AGNR. In a real device, the relaxation
distance between the top and bottom of the potential well depends on the
thickness and dielectric constant of the substrates. The thinner the
insulators (e.g., hBN) are, the sharper the potential well will be. The on-
site potential of carbon atoms in a real 2D device is shown in Fig. 1(c), in
which the YJ-GWG formed by the combination of two S-shape waveguides, is
indicated by dashed lines. The 2D device (AGNR) consists of two main parts.
First, a fairly large intrinsic channel with width W and Length L, while
$W_{sa,da1,da2}$ represent the width of three arms of YJ-GWG, i.e., source
arm, lower drain arm and upper drain arm, respectively. In the standard
convention, AGNRs are labeled as NA-AGNR where NA is the number of dimer lines
defined by N${}_{A}=\lfloor{W/(0.5\sqrt{3}a_{cc})}\rfloor$, W is the width of
AGNR and $a_{cc}=0.142$ nm is the carbon-carbon bond length. The second part
of our device is the contacts (source and drains) which are also made of
carbon and are in fact extensions from the scattering area, as illustrated by
vertical red lines in Fig. 1(c). The widths of all terminals are tailored to
be matched with waveguide arms, i.e., WS = Wsa, W${}_{D_{1,2}}$ =
W${}_{da_{1,2}}$. In all numerical examples, we have chosen metallic type of
supercells (or (3m+2) family of AGNRs) for all terminals to artificially
replicate a real metal contact. The reason to use these GNRs instead of metal
is to provide good ohmic contacts owing to the workfunction match. The length
of scattering area (L) is related to the number of supercells (NS) via
N${}_{S}=\lfloor{L/(3a_{cc})}\rfloor$. WO represents the amount of overlap
between the two flat ends of S-shape GWGs, and $W_{MB}$ is the width of middle
barrier between the two $drain~{}arm$s. Wsa and WMB are related to WO via
Wsa=W${}_{da_{1}}$+W${}_{da_{2}}$-WO and WO/2+WMB/2=Wbe, where Wbe can be
viewed as the amount of bending.

###  II.2 Numerical model

The $p_{z}$ orbital wavefunction of Carbon is used to construct TB-Hamiltonian
for channels and terminals,

$H=\sum_{\langle{ij}\rangle}-t_{ij}a_{i}^{\dagger}a_{j}+\sum_{i}v_{i}a_{i}^{\dagger}a_{i},$
(1)

where $a_{i}^{\dagger}$ and $a_{i}$ are the creation and annihilation
operators at $i$-th atomic site, respectively. The symbol
${\langle{ij}\rangle}$ denotes a pair of atomic sites and the relevant hopping
parameters $t_{ij}$ are real energy values.

Table 1: TB parameter sets. Set | $E_{2p}(eV)$ | $t_{1}(eV)$ | $t_{2}(eV)$ | $t_{3}(eV)$ | $s_{1}$ | $s_{2}$ | $s_{3}$
---|---|---|---|---|---|---|---
1NN | 0 | -2.78 | 0 | 0 | 0 | 0 | 0
3NNI | -0.45 | -2.78 | -0.15 | -0.095 | 0.117 | 0.004 | 0.002
3NNII | -0.187 | -2.756 | -0.071 | -0.38 | 0.093 | 0.079 | 0.070

We have assumed the first nearest and two other third nearest approximations
denoted by 1NN and 3NNI,II in different cases to comprehend the impact of them
on transport. The relevant TB parameters for 1NN and 3NNI,II are shown in
Table 1. In particular, the new TB parameters, 3NNII, obtained from density
functional theory (DFT), has shown excellent agreement with the predictions of
the DFT even in the high-energy region reich2002tight ; kundu2011tight ;
tran2017third . The effect of the potential well on the transmission is
directly considered through the on-site potential energy on the Hamiltonian,
i.e., $v_{i}$ in Eq. (1). The assigned potential map $U$ for the Y-junction is
shown in Fig. 1(c), in which the value of $v_{i}$ in the bottom of Y-shape
potential well is $u_{in}$, and that in the barrier regions has a fix value
$u_{out}=0$. Similarly, on-site potential of source and drains terminals in
their Hamiltonian are noted as $u_{S}$ and $u_{D_{1,2}}$, which are used to
represent an applied bias voltage. The effect of edge roughness in the GNR is
not included in our calculation gunlycke2008tight . Because the edge roughness
can be far from the potential-defined Y-junction and if a very wide GNR is
considered, their effect on the transport in YJ-GWG can be neglected. The
effect of electron-phonon interactions is not included in this work either,
due to the weak electron-phonon interactions in graphene borysenko2010first .
Using the Landauer-Bütticker formalism datta2005quantum , the conductance
between the source and drains in low temperature can be expressed as:

${\rm G_{SD_{1,2}}(E)}={\rm G_{0}}{\rm Tr}[{\Gamma_{\rm S}(E)}{G^{\rm
r}(E)}{\Gamma_{\rm D_{1,2}}(E)}{G^{\rm a}(E)}],$ (2)

where ${\rm G_{0}}=2e^{2}/h$ is the quanta of conductance that includes the
spin degree of freedom, $\Gamma_{\rm S,~{}D_{1,2}}$ are the source and drains
broadening matrices, which couple the left and right terminals to the central
scattering region in YJ-GWG. The retarded Green’s function $G^{\rm r}$ is
defined as ${G^{\rm r}(E)}=[(E+i\eta)S-H-\Sigma_{\rm S}(E)-\Sigma_{\rm
D_{1,2}}(E)]^{-1}$ and $G^{\rm a}=(G^{\rm r})^{\dagger}$, where S is the
overlap matrix and is in the form of the first part in equation (1). The open
boundary condition of terminals is added via $\Gamma_{\rm
S,D_{1,2}}(E)=i(\Sigma_{\rm S,D_{1,2}}^{\rm r}(E)-\Sigma_{\rm S,D_{1,2}}^{\rm
a}(E))$, where $\Sigma_{\rm S,D_{1,2}}^{\rm a}=(\Sigma_{\rm S,D_{1,2}}^{\rm
r})^{\dagger}$ and $\Sigma_{\rm S,D_{1,2}}^{\rm r}$ are self-energies of the
semi-infinite terminals on the source and drains. The transport simulation of
YJ-GWG is carried out by our ballistic code in which we have used the direct
recursive algorithm to calculate conductance of a system with few hundred
thousand of atoms lewenkopf2013recursive ; thor2014recursive . This Memory-
friendly approach allows us to perform partial inversion of a gigantic matrix,
which is a necessary mathematical operation to calculate the Green’s function
in real space.

##  III Results and Discussion

###  III.1 Tunning

Y-junction gate voltage is used to control the depth of quantum well and
consequently provide deep-enough confined energy channels. By this manner, the
bottom of on-site potential, i.e., $u_{in}$, can be parametrically tuned to
achieve a coherent transport in the device. Moreover, our previous work has
demonstrated that on-site potential of terminals is an equally vital parameter
to achieve the coherent transport in a two-terminal graphene waveguide
mosallanejad2017perfectly .

Figure 2:  Conductances of two-terminal curved S-GWGs in various $u_{in}$.
Right (left) panels show the tunning of a S-GWG with the waveguide width
$W_{D}$ of 10 nm (20 nm).

It has been concluded that the quantization of conductance occurs if the
following conditions were met: (i) the terminals have similar width of the
waveguide, (ii) the source has a fix on-site potential almost equal to that in
the bottom of waveguide (i.e., $u_{S}=u_{in}$), (iii) on-site potential at the
drain is kept zero, which is equal to $u_{out}$.To quantitatively determine an
appropriate range of $u_{in}$ as a primary study, a few two-terminal transport
studies were performed for two curved S-GWGs with waveguide widths ($W_{D}$)
of 10 nm and 20 nm. The on-site potential profile of such systems is similar
to that in Fig. 1(c) but the negative potential well (blue color) is placed
only within one arm. For each sample, $u_{in}$ has been swept over a range of
values under the conditions of $u_{S}=u_{in}$ and $u_{D_{1,2}}=0$.
Conductances on all upper panels of Fig. 2 (in both sizes), show that
shallower wells (small negative $u_{in}$) are incapable to produce smooth
plateaus (note that we have employed 1NN approximation in these calculations).
Once the well gets deeper, more well-defined smooth plateaus start to form.
Furthermore, narrower waveguide (10 nm) needs deeper potential well to produce
smooth plateaus. Thus, it is essential to have a deep-enough well via applying
appropriate gate voltages. Independent comparison between the right and left
panels of Fig. 2 shows that number of plateaus within the same energy range is
somehow, insensitive to the depth of potential well but rather dependent on
the width of the waveguide. This fact indicates that the width of waveguide
determines the conductance profile if the potential is deep enough. Since
$u_{in}=-0.30$ provide smooth plateaus for the 10 nm wide waveguide,
hereafter, we will use these quantitative values in the transport study of YJ-
GWG in the next section.

###  III.2 Transport study in YJ-GWG

Start with a given drain width $W_{D_{1},D_{2}}=10$ nm, four samples with
different $W_{O}$ and $W_{S}$ are created and the conductances of both paths
in each sample are evaluated. The dimension of the system is $W\times
L=80\times 120~{}nm^{2}$ and paths have symmetry around $y=W/2$. Center of
bending is located in the middle of x-direction at $L/2$. Calculated results
are shown in Fig. 3(a)-(d).

Figure 3:  Conductances of both paths in a smooth/S-shape YJ-GWG. Dot and
solid lines denote the conductances of lower and upper paths receptively.
(a)-(d) are the conductances for different values of $W_{O}$ as it is noted.

$W_{O}$ has been selected as the first control parameter to study the effect
of overlap, partially because other parameters, such as $W_{be}$, (which
represents the amount of bending) has shown very less influences on the
conductance of a S-GWG. Lower panels of Fig. 3, (c) and (d), with smooth and
identical plateaus for both paths, demonstrate that it is possible to
transport carriers coherently in both paths of a YJ-GWG provided that the
parameter $W_{O}$ is small enough. By comparing all panels in Fig. 3, one can
find a noticeable smooth conductance which was achieved when the width of the
source arm increases (because of Wsa=W${}_{da_{1}}$+W${}_{da_{2}}$-WO). In
addition, another type of YJ-GWG has been constructed based on a kink-shape
bending to study the effect of sharp bending. Variation of waveguide width
occurs around the sharp bending, indicated by arrows in the Fig. 4(a), within
which the width of waveguide differs from $W_{da1,da2}$ in comparison with the
fully-smooth S-shape bending shown by dashed lines in the Fig. 4(a).

Figure 4:  (a) Borders with which potential well of the smooth and non-smooth
YJ-GWG have been defined. Dashed lines refer to smooth or S-shape bending
whereas the solid lines refer to non-smooth or kink-shape bending. (b)
Equivalent potential profile of the non-smooth Y-junction.

S-shape bending ensures that the width of both paths remains unchanged after
the splitting point. Since the energy channels are highly sensitive to the
width of single-waveguide, width variations in sharp bendings are expected to
induce extra scatterings that affect the conductance quality. Fig. 5(a)-(d)
show the calculated conductances of the kink-shape YJ-GWG with different
$W_{O}$. Fig. 5 supports our argument and indeed shows more noisy
conductances, especially in the second plateau.

Figure 5:  Conductances of both paths in a non-smooth Y-junction graphene
waveguide. Dot and solid lines denote the conductances of lower and upper
paths receptively. (a)-(d) are the conductances for different values of
$W_{O}$ as it is noted.

Note that except for the shape of the waveguide, all other parameters used in
Fig. 5 are similar to those used in Fig. 3 (note that the third nearest
approximation, denoted by 3NNI has been employed for both cases).

We further have investigated the geometrical effect on transport by enlarging
the whole device by a scale factor of 1.5 but keeping the Y-shape structure
unchanged. Accordingly, scattering area of $W\times L=120\times 180~{}nm^{2}$,
arm’s width of $W_{da_{1},da_{2}}=15$ nm, $W_{O}=3$ nm and
$u_{S}=u_{in}=-0.25~{}(eV)$ have been taken into the account.

Figure 6:  (a) Color lines refer to the conductances of both paths of a YJ-GWG
scaled by factor 1.5, employing energy set of 3NNI. (b) Color lines refer to
the conductances of same configuration utilizing energy set of 3NNII from
Table. 1. Red-hallow circles refers to the two terminal conductance of the
relevant S-GWG employing 3NNI.

Conductances of two paths have been shown by dashed and solid lines in Fig.
6(a). Comparison of conductances between Fig. 3(d) and Fig. 6(a) implies that
scaling has minor effect on general characterizations of the YJ-GWG. In Fig.
6(a), the third nearest approximation, denoted by 3NNI has been used whereas
3NNII is employed to calculate conductances of YJ-GWG in Fig. 6(b). One can
conclude that the new TB parameter set has lifted the valley degeneracy and
produced multiple sub-plateaus illustrated by arrows in Fig. 6(b). Careful
observers may notice seminal plateaus on the results calculated by 3NNI as
well [Fig. 6(a)]. Red-hallow circles show the conductance of the relevant two-
terminal S-GWG (only one arm) with similar parameters and 3NNI energy set as a
reference. Comparison between red-hallow circles, dashed and solid lines in
Fig. 6(a) tells us, the conductances of YJ-GWG almost follow the conductance
of a relevant S-GWG. In Fig. 6(b), it also shows 3NNII approximation results
in a shift of conductances toward the positive side of energy axis.

###  III.3 Bandstructure Remarks

In order to obtain more insight on our results, we have calculated quasi-one
dimensional bandstructure for two supercells, one at the left end of AGNR,
e.g., x = 10 nm (with single-well) and the other at the right end, e.g., x =
90 nm (with double well). The positions of those supercells are specified by
blue and red dots in Fig. 1(b). The calculated bands, which contain sixteen
conduction bands and one valance band in the positive momentum space, are
shown in Fig. 7(a)-(d). Thus, it is possible to track the movement of subbands
in energy space at different $W_{O}$. A single-well possess two-fold
degenerate bands which are highlighted by blue dashed and black narrow solid
lines whereas double-well results in four-fold degenerate bands which are
denoted by a red dashed, a red dotted line and two (multi-color) narrow solid
lines together. Subbands belong to the barriers, which are located above the
highlighted bands, are shown by multi-color narrow solid lines. All panels in
Fig. 7 show that change in $W_{O}$ (or $W_{MB}$) does not shift the energy
bands of double-well substantially, whereas similar change of $W_{O}$ shift
down the bands of a single well. Note that decrease in $W_{O}$ leads to a
similar increase in $W_{MB}$, because $W_{O}/2+W_{MB}/2=W_{be}$, where
$W_{be}$ is fixed. Thus adjusting $W_{O}$ is a practical method to match
energy channels at the two ends of scattering area. This diagram also explains
why the smooth conductance occurs when $W_{O}=4$ nm at Fig 3(c). We can see
the two types of energy bands, shown by a tip of arrow, cross each other
around the Dirac point at Fig. 7(c). The calculated band structures indicate
that lower energy bands with $E<0$ do not participate in the quantized
transport.

Figure 7:  (Color plot) Quasi-one dimensional bandstructure of supercells that
are extracted from the proposed YJ-GWG, Fig. 1(b). A blue dashed and a solid
line show two-fold degenerate bands of a AGNR’s supercell at the left side of
device in presence of single well, whereas a red dashed,a red dotted and two
solid lines together show four-fold degenerate subbands of a AGNR supercell at
the right side of YJ-GWG where two wells are separated by $W_{MB}$. 3NNI is
employed.

It is possible to explain a physical reason of our theoretical observation
from the energy point of view. It is widely accepted that scattering happens
as a result of mismatch between energy levels along the transport direction.
For example, impurity levels might destructively couple to transport channels
and smear the quantization of conductance. Here in YJ-GWG, one can expect
minimum scattering if energy channels merge to each other with minimum
misalignment around Y-junction. Therefore, we have further investigated our
study by extracting spatial-resolved energy spectrum, i.e. waveguide energy
channels, of few super-cells along x and around the splitting point, as
plotted in Fig. 8(a) and (b) for a fix momentum kx= 0.405 ($nm^{-1}$).
Relative on-site potential used for the calculation in Fig.8 (a) and (b) are
plotted in Fig. 8(c) and (d), respectively. One can see that two-fold
degenerate energy bands (second and fourth levels) on the left side of
Y-junction merge to four-fold degenerate bands on the right side. The amount
of energy level variations are much less for the case of Wo = 2nm as compare
to Wo = 8nm. Note that, one can expect similar merging trends for other higher
momentums, since the E-kx relationship is linear for other larger values of
momentum. From band structure studies, Fig. 8(a) and (b), one can see
obviously that the energy spacing between first and second four-fold bands,
spatially available on the right side of device, is about 150 meV which is
noticeably large and consistent with the width of first flat plateaus on the
energy range at Fig. 3.

Figure 8:  Spatial-resolved quasi-one dimensional energy spectrum for smooth
YJ-GWG around splitting point. (a) $W_{o}$=8 nm. (b) $W_{o}$=2 nm. (c) On-site
potential for selected supercells of (a). (d) On-site potential for selected
supercells of (b).

###  III.4 Discussion

After the realization of quantized conductance on quantum point contact (QPC),
Y-junction structures on modulation-doped heterostructures have been proposed
and fabricatedvan1988quantized ; maao1994quantum ; shorubalko2003tunable .
Different terminologies have been used to address similar devices such as
Y-branch switch (YBS) and three-terminal ballistic junction (TBJ). A gate
defined YBS hardly works in ballistic regime or in another world the
quantization of conductance is rarely reported as a prominent characteristic
of deviceshorubalko2003tunable . Whereas, peaks on the derivative of
conductance on wet-etched version of YBS demonstrate the presence of
quantizationworschech2005nonlinear ; jones2005quantum . This is probably
because subband-spacings in a gate defined 1D quantum wire are limited to the
order of a few meV while larger energy spacing in a wet-etched GaAs/AlGaAs
quantum wires, typically on the order of a few tens of meV, have been
reported. Note that, the conductance of wet-etched YBS does not exhibits flat
plateaus and an exact identical I-V characteristic have not been reported. On
the other hand, carbon nanotube Y-junction (CNT-YJ), T-shaped GNR (T-GNR) and
few Cross-shape GNR (C-GNR) have been widely studied
theoreticallyandriotis2001rectification ; andriotis2002transport . The
conductance of CNT-YJ shows oscillatory behavior in theoretical studies which
merely depends on geometry of system. To the best our knowledge quantization
of conductance does not report on chemically synthesized Y-junctions nanotube
and both T-GNR and C-GNR do not exhibit quantization of conductance as
wellouyang2009transport ; bandaru2005novel ; brandimarte2017tunable . A
prominent benefit of the proposed structure is the flat plateaus on
conductance of a smooth YJ-GWG, Fig. 3(d), which is not reported in other
possible carbon base splitters so far. The optimization of the width of
incoming path on a smooth-bended YJ-GWG can be considered as a viable method
to achieve the goal of coherent splitting. YJ-GWG can be a gateway for the
realization of possible carbon-base electron interference devices, as well as
interconnect in graphene-based spintronic devices.

##  IV Conclusions

In conclusion, we have exploited the possibility of coherent transport through
a Y-junction graphene waveguide. Results show that if the incoming path of YJ-
GWG is wide enough, the conductances of both paths are identical to the single
graphene waveguide. It has been shown that smoothly bended GWG provide much
smoother plateaus on the conductance. The primitive YJ-GWG enlarged by a scale
factor of 1.5 and the results of transport study show that the quantization of
conductance is well preserved in a larger device. In addition, we have shown
that the influence of different TB energy sets on the calculation of
conductance. A quasi-one dimensional bandstructure calculation was also
performed to explain our observations.

## Acknowledgments

This work was supported by the National Key Research and Development Program
of China (Grant No. 2016YFA0301700), the National Natural Science Foundation
of China (Grants No. 11625419），and the Anhui Initiative in Quantum Information
Technologies (Grants No. AHY080000). We should also thank Chinese Academy of
Sciences and The World Academy of Science for the advancement of science in
developing countries.

##  Appendix A Y-junction on-site potential

We have accomplished the following steps to build a 2D on-site potential of
YJ-GWG including relaxation distance where values of potential energy inside
and outside of Y-junction are uin and uout.
Step 1 ; Calculate functions, $Y_{du}(x_{i})$ and $Y_{dl}(x_{i})$, to define
upper and lower edge lines of the downward bended S-GWG (i.e., downward lines
in Fig. 9(a)),

$\displaystyle Y_{du}(x_{i})$ $\displaystyle=$
$\displaystyle(Y_{0}+W_{O}/2)+W_{be}(f(x_{i})-1)$ (3) $\displaystyle
Y_{dl}(x_{i})$ $\displaystyle=$ $\displaystyle Y_{du}(x_{i})-W_{da2}$ (4)

and $Y_{ul}(x_{i})$ and $Y_{uu}(x_{i})$ to define lower and upper borders of
the upward bended S-GWG (i.e., upward lines in Fig. 9(a))

$\displaystyle Y_{ul}(x_{i})$ $\displaystyle=$
$\displaystyle(Y_{0}-W_{O}/2)+W_{be}(1-f(x_{i}))$ (5) $\displaystyle
Y_{uu}(x_{i})$ $\displaystyle=$ $\displaystyle Y_{ul}(x_{i})+W_{da1}$ (6)

where $f(x_{i})=1/(1+exp((x_{i}-x_{0})/\sigma_{x})$ is the Fermi function
which is employed only to produce smooth variation along the x direction.
$Y_{0}=W/2$ sets the center of YJ-GWG on the y direction, $x_{0}$ is the
inflection point of bended lines on the x direction and $\sigma_{x}$ defines
the sharpness of bending in this direction. A bigger $\sigma_{x}$ provides a
smooth and adiabatic variation along the x direction. Throughout all studies
with smooth bending, the value of $\sigma_{x}$ is $L/14$.
Step 2 ; Once again Fermi function is employed to establish another smooth
step-like functions in y-direction which their inflection points define by
making use of previously determined border functions i.e.,
$Y_{ul,uu,du,dl}(x_{i})$ as follow;

$\displaystyle F_{bo}(x_{i},y_{i})=1/(1+exp((y_{i}-Y_{bo}(x_{i}))/\sigma_{y})$
(7)

where the index $bo$ refers to one of ul, uu, du and dl. We should remind that
$x_{i}$ and $y_{i}$ are x and y coordinations of i-th atom. $\sigma_{y}$
determines relaxation distance ($\Delta W$) on transverse direction. One can
easily conclude $\Delta W=2\sigma_{y}Ln(9)\approx 4.4\sigma_{y}$ if
$F_{bo}=0.9$ and $F_{bo}=0.1$ approximate upper and lower limits of Fbo. We
have considered $\Delta W\approx$4 nm on all of our structures.
Step 3 ; The subtraction $F_{out}=(F_{uu}-F_{dl})$ provide a function which
smoothly varies from zero on both side-barrier areas to the one on the middle
of outer lines i.e., $Y_{uu}$ and $Y_{dl}$, red solid lines in Fig. 9(a).
$F_{out}$ is plotted in Fig. 9(b). Utilization of Fermi function on step 2
ensures that the full width of half maximum (FWHM) of waveguide is equal to
the distance between border lines because the value of $F_{bo}$s are 1/2 on
inflection points. Note that $Y_{du}$ and $Y_{ul}$, dashed lines in Fig. 9(a),
cross each other on the splitting point ($Y_{0},x_{sp}$). At the same time,
for $x_{i}>x_{sp}$ the subtraction $F_{in}=(F_{du}-F_{ul})$ provide us a
function which smoothly varies from negative one on the middle-barrier (MB)
region, the areas that is trapped between inner border lines (du and ul), to
zero on side-barrier areas, see Fig. 9(c). $F_{in}$ gives a positive value for
overlapping area $x_{i}<x_{sp}$ which we cancel it by signing the positive
values to zero and call the new function $F_{in-ne}$.

Figure 9:  (color online) Steps to build an on-site potential for YJ-GWG with
smooth relaxation. (a) Border lines with which two bended S-GWG can be
defined. (b) $F_{out}$. (c) $F_{in}$. Note $F_{in-ne}$ refers to $F_{in}$ on
the right side of xsp. (d) $F_{out}+F_{in-ne}$.

Step 4 ; $F_{out}+F_{in-ne}$ delivers a Y-splitter function with zero value on
all (three) barriers and positive one within the Y-branch, see Fig. 9(d).
Step 5 ; Finally, proper estimation of on-site potential of YJ-GWG is given by
adding potential amplitudes as follow

$\displaystyle v_{i}(x_{i},y_{i})=u_{out}+(u_{in}-u_{out})[F_{out}+F_{in-
ne}].$ (8)

One can build a kink-shape on-site potential with similar process except kink-
shape border lines must be taken into account instead of smooth lines in step
1.

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