# Continuous Edge Chromatic Numbers of Abelian Group Actions

Su Gao  School of Mathematical Sciences and LPMC, Nankai University, Tianjin
300071, P.R. China<EMAIL_ADDRESS>,  Ruijun Wang  School of Mathematical
Sciences and LPMC, Nankai University, Tianjin 300071, P.R. China
<EMAIL_ADDRESS>and  Tianhao Wang  School of Mathematical Sciences
and LPMC, Nankai University, Tianjin 300071, P.R. China<EMAIL_ADDRESS>
###### Abstract.

We prove that for any generating set $S$ of $\Gamma=\mathbb{Z}^{n}$, the
continuous edge chromatic number of the Schreier graph of the Bernoulli shift
action $G=F(S,2^{\Gamma})$ is $\chi^{\prime}_{c}(G)=\chi^{\prime}(G)+1$. In
particular, for the standard generating set, the continuous edge chromatic
number of $F(2^{\mathbb{Z}^{n}})$ is $2n+1$.

###### Key words and phrases:

Cayley graph, Schreier graph, Bernoulli shift action, edge chromatic number,
marker region lemma

###### 2020 Mathematics Subject Classification:

Primary 03E15; Secondary 05C15, 05C70

The first author acknowledges the partial support of his research by the
National Natural Science Foundation of China (NSFC) grants 12250710128 and
12271263.

##  1\. Introduction

In the seminal paper [12], Kechris, Solecki and Todorcevic initiated the study
of descriptive combinatorics of locally finite Borel graphs. In particular,
they considered general bounds for Borel chromatic numbers of locally finite
Borel graphs. The Borel edge chromatic number was also considered, primarily
as the Borel chromatic number of the dual graph, and they showed that any
locally finite Borel graph $G$ has countable Borel edge chromatic number
$\chi^{\prime}_{B}(G)$, and if $G$ is of bounded degree $\leq k$, then
$\chi^{\prime}_{B}(G)\leq 2k-1$. Subsequently, researchers computed the Borel
chromatic numbers and Borel edge chromatic numbers of various Schreier graphs
of countable group actions. For example, in [13], Marks studied combinatorics
of free products of two marked groups, and showed that the Borel chromatic
number $\chi_{B}(F(2^{\mathbb{F}_{n}}))=2n+1$, where $F(2^{\mathbb{F}_{n}})$
denotes the Schreier graph on the free part of the Bernoulli shift action of
$\mathbb{F}_{n}$, the free group with $n$ generators, on $2^{\mathbb{F}_{n}}$.
For an overview of the entire field of descriptive combinatorics we refer the
reader to the survey [11] by Kechris and Marks.

The continuous chromatic number was first studied by the first author of the
present paper and his coauthors. In [6], it was shown that there is a
continuous proper $4$-coloring of $F(2^{\mathbb{Z}^{n}})$ for each $n>1$.
Later in [7], it was shown that there is no continuous proper $3$-coloring of
$F(2^{\mathbb{Z}^{n}})$ for each $n>1$. Thus the continuous chromatic number
$\chi_{c}(F(2^{\mathbb{Z}^{n}}))=4$ for each $n>1$. In contrast, it was shown
in [3] and [8] by different methods that the Borel chromatic number
$\chi_{B}(F(2^{\mathbb{Z}^{n}}))=3$ for each $n>1$. This was the first time
that the Borel and continuous chromatic numbers of a locally finite Borel
graph were observed to be different.

A similar phenomenon occurs when considering the Borel and continuous edge
chromatic numbers of $F(2^{\mathbb{Z}^{n}})$. The Borel edge chromatic numbers
were studied independently in [1] (for $n=2$), [2], [9] and [14]. The
conclusion is that for the Schreier graph of $F(2^{\mathbb{Z}^{n}})$ with the
standard generating set, the Borel edge chromatic number
$\chi^{\prime}_{B}(F(2^{\mathbb{Z}^{n}}))=\chi^{\prime}(F(2^{\mathbb{Z}^{n}}))=2n$.
It is worth noting that Weilacher [14] showed that for any symmetric
generating set $S$ of $\mathbb{Z}^{n}$ where $S$ does not contain the
identity, letting $F(S,2^{\mathbb{Z}^{n}})$ be the corresponding Schreier
graph, we have
$\chi^{\prime}_{B}(F(S,2^{\mathbb{Z}^{n}}))=\chi^{\prime}(F(S,2^{\mathbb{Z}^{n}}))=|S|$.
Thus the computation of the Borel edge chromatic numbers of
$F(2^{\mathbb{Z}^{n}})$ was complete. As for the continuous edge chromatic
number, we only had, from [7], the result for $n=2$ that
$\chi^{\prime}_{c}(F(2^{\mathbb{Z}^{2}}))=5$ for the standard generating set.
This already showed that the Borel and continuous edge chromatic numbers could
differ, but left the computation of a lot of cases open.

In this paper, we complete this line of research by showing that, for any
symmetric generating set $S$ of $\mathbb{Z}^{n}$ not containing the identity,
the continuous edge chromatic number
$\chi^{\prime}_{c}(F(S,2^{\mathbb{Z}^{n}}))=\chi^{\prime}(F(S,2^{\mathbb{Z}^{n}}))+1=|S|+1$.
In particular, we have $\chi^{\prime}_{c}(F(2^{\mathbb{Z}^{n}}))=2n+1$. This
answers Question 10.3 of [5].

######  Theorem 1.1.

For any integer $n\geq 1$ and any symmetric generating set $S$ of
$\mathbb{Z}^{n}$ not containing the identity, let $G=F(S,2^{\Gamma})$. Then
$\chi^{\prime}_{c}(G)=\chi^{\prime}(G)+1=|S|+1$.

The rest of the paper is organized as follows. In Section 2 we fix the
notation to be used throughout the paper. In Section 3 we prove the lower
bound for the continuous edge chromatic number. Then in Section 4 we construct
a proper edge coloring to witness Theorem 1.1, first by assuming $S$ to be the
standard generating set and then generalizing to the general case.

##  2\. Preliminaries

In this section we define our basic notions and fix notation.

For a graph $G$ and a set $\kappa$ of colors, a proper edge $\kappa$-coloring
of $G$ is a map $c:E(G)\to\kappa$ such that $c((x,y))\neq c((z,w))$ if the
distinct edges $(x,y),(z,w)$ have a vertex in common. The edge chromatic
number of $G$, denoted by $\chi^{\prime}(G)$, is the least cardinality of a
set $\kappa$ such that there exists a proper edge $\kappa$-coloring for $G$.
When the graph $G$ is a topological graph, i.e., when $V(G)$ is a topological
space, we may consider continuous (or Borel) proper edge colorings of $G$, and
define its continuous edge chromatic number, denoted as
$\chi^{\prime}_{c}(G)$, and its Borel edge chromatic number, denoted as
$\chi^{\prime}_{B}(G)$. Similarly, we define proper (vertex) colorings and the
continuous and Borel chromatic numbers, denoted by $\chi_{c}(G)$ and
$\chi_{B}(G)$ respectively.

On any locally finite graph $G$, we define the path distance $\rho$ on $G$ as
follows. If $x,y\in V(G)$ are in the same connected component (i.e. there is a
path from $x$ to $y$), then let $\rho(x,y)$ be the length of a shortest path
from $x$ to $y$; otherwise let $\rho(x,y)=\infty$. Recall that a locally
finite graph $G$ is regular if the degrees of all vertices of $G$ are the
same.

An important class of regular topological graphs consists of the graphs
induced by actions of finitely generated groups. A marked group is a pair
$(\Gamma,S)$, where $\Gamma$ is a group and $S$ is a finite generating set of
$\Gamma$. Usually we also require $S=S^{-1}$ and $1_{\Gamma}\not\in S$. In
this paper this is assumed tacitly. When the generating set is standard or
otherwise understood, we omit specifying the set $S$ and say that $\Gamma$ is
a marked group. For example, for any $n\geq 1$, $\mathbb{Z}^{n}$ has a
standard generating set $\\{\pm e_{i}\colon 1\leq i\leq n\\}$ where each
$e_{i}$ has the $i$-th coordinate $1$ and $j$-th coordinate $0$ for $j\neq i$.
The Cayley graph $C(\Gamma,S)$ of a marked group $(\Gamma,S)$ is defined by

$V(C(\Gamma,S))=\Gamma$

and

$E(C(\Gamma,S))=\\{(g,h)\in\Gamma^{2}\,:\,\exists s\in S\ h=sg\\}.$

When there is an action of a marked group $\Gamma$ on a space $X$, the
Schreier graph of the action $\Gamma\curvearrowright X$ on $X$, denoted
$G(\Gamma,S,X)$, is defined by

$V(G(\Gamma,S,X))=X$

and

$E(G(\Gamma,S,X))=\\{(x,y)\in X^{2}\,:\,\exists s\in S\ y=s\cdot x\\}.$

The Schreier graph will be particularly nice when the action is free; in this
case the Schreier graph on each orbit of the action will be a copy of the
Cayley graph $C(\Gamma,S)$. Let $F(\Gamma,X)=\\{x\in X\colon\forall g\neq
1_{\Gamma}\ (g\cdot x\neq x)\\}$ be the free part of the action
$\Gamma\curvearrowright X$. $F(\Gamma,X)$ is an invariant set and the induced
action of $\Gamma$ on $F(\Gamma,X)$ is free. We denote by $F(\Gamma,S,X)$ the
Schreier graph on $F(\Gamma,X)$. Oftentimes, we write $F(X)$ for $F(\Gamma,X)$
and $F(S,X)$ for $F(\Gamma,S,X)$ if the group $\Gamma$ and its action on $X$
are understood. Furthermore, when the generating set $S$ is standard or
otherwise understood, we will abuse notation and simply write $F(X)$ for the
Schreier graph $F(S,X)$.

In this paper, we will work with the setup where $X$ is a Polish space, i.e.,
a separable and completely metrizable space, $\Gamma$ is a finitely generated
group, and the action $\Gamma\curvearrowright X$ is continuous.

The Bernoulli shift action induces a natural Schreier graph. Let $A$ be a
finite set with the discrete topology. For a countable group $\Gamma$, the
space $A^{\Gamma}$ is equipped with the usual product topology and is
homeomorphic to the Cantor space $2^{\mathbb{N}}=\\{0,1\\}^{\mathbb{N}}$,
which is compact Polish. The Bernoulli shift action $\Gamma\curvearrowright
A^{\Gamma}$ is defined by

$(g\cdot x)(h)=x(hg)$

for $g,h\in\Gamma$ and $x\in A^{\Gamma}$. This action is continuous. The free
part $F(\Gamma,A^{\Gamma})$ is an invariant dense $G_{\delta}$ subset of
$A^{\Gamma}$, hence is a Polish space.

We are now ready to compute the continuous edge chromatic number
$\chi^{\prime}_{c}(F(S,2^{\mathbb{Z}^{n}}))$ for a generating set $S$ of
$\mathbb{Z}^{n}$.

##  3\. The Lower Bound

In this section we prove that for any generating set $S$ of $\mathbb{Z}^{n}$,
$\chi^{\prime}_{c}(F(S,2^{\mathbb{Z}^{n}}))>\chi^{\prime}(F(S,2^{\mathbb{Z}^{n}}))$.
We will need some results of [7]. To state the results we need to review some
related concepts.

Let $(\Gamma,S)$ be a marked group. We will consider the Bernoulli shift
action of $\Gamma$ on $A^{\Gamma}$ for some finite set $A$. A
$\Gamma$-subshift $Y$ is a closed invariant subset of $A^{\Gamma}$. A
$\Gamma$-pattern is a map $p:F\to A$ for some finite $F\subseteq\Gamma$. If
$p$ is a $\Gamma$-pattern and $x\in A^{\Gamma}$, we say that $p$ occurs in $x$
if $p\subseteq g\cdot x$ for some $g\in\Gamma$. Note that this is an invariant
notion, that is, if $p$ occurs in $x$ then $p$ occurs in any $h\cdot x$ for
$h\in\Gamma$. A $\Gamma$-subshift of finite type is a $\Gamma$-subshift
$Y\subseteq A^{\Gamma}$ for which there is a finite set
$\\{p_{1},\dots,p_{k}\\}$ of $\Gamma$-patterns such that for any $x\in
A^{\Gamma}$, $x\in Y$ if and only if none of the patterns $p_{1},\dots,p_{k}$
occur in $x$. In this case we say that $Y$ is described by
$(A;p_{1},\dots,p_{k})$.

A $(\Gamma,S)$-graph $G$ is a graph with an action of $\Gamma$ on $G$ such
that for any $x,y\in V(G)$, $(x,y)\in E(G)$ if and only if there is $g\in S$
such that $g\cdot x=y$. The Schreier graphs $G(\Gamma,S,X)$ and
$F(\Gamma,S,X)$ are examples of $(\Gamma,S)$-graphs, but we will also consider
finite $(\Gamma,S)$-graphs. When the generating set $S$ is standard,
$(\Gamma,S)$-graphs are also simply called $\Gamma$-graphs.

We will work with the following example of a finite $\mathbb{Z}^{n}$-graph.
Given positive integers $q_{1},\dots,q_{n}$, the vertex set of the graph
$T_{q_{1},\dots,q_{n}}$ is the quotient of $\mathbb{Z}^{n}$ by the relation
$(a_{1},\dots,a_{n})\sim(b_{1},\dots,b_{n})$ if and only if $a_{1}\equiv
b_{1}\\!\mod q_{1}$, $\dots,a_{n}\equiv b_{n}\\!\mod q_{n}$. The group
$\mathbb{Z}^{n}$ acts naturally on $T_{q_{1},\dots,q_{n}}$, and this action
induces a $\mathbb{Z}^{n}$-graph on $T_{q_{1},\dots,q_{n}}$ with the standard
generating set of $\mathbb{Z}^{n}$. Note that $T_{q_{1},\dots,q_{n}}$ has
cardinality $q_{1}\cdots q_{n}$.

If $Y$ is a $\mathbb{Z}^{n}$-subshift of finite type described by
$(A;p_{1},\dots p_{k})$, then we say that a map $\varphi\colon
T_{q_{1},\dots,q_{n}}\to A$ respects $Y$ if none of $p_{1},\dots,p_{k}$ occur
in $\varphi\circ\pi\in A^{\mathbb{Z}^{n}}$, where $\pi\colon\mathbb{Z}^{n}\to
T_{q_{1},\dots,q_{n}}$ is the quotient map.

If a group $\Gamma$ acts on both $X$ and $Y$, we say a map $\theta\colon X\to
Y$ is equivariant if $\theta(g\cdot x)=g\cdot\theta(x)$ for all $g\in\Gamma$
and $x\in X$.

######  Theorem 3.1.

([7, Theorem 2.7.1]) Let $A$ be a finite set and let $Y\subseteq
A^{\mathbb{Z}^{n}}$ be a $\mathbb{Z}^{n}$-subshift of finite type. If there is
a continuous equivariant map $\theta\colon F(2^{\mathbb{Z}^{n}})\to Y$, then
for all $q_{1},\dots,q_{n}\geq 2$ and all sufficiently large $k$ there is
$\varphi\colon T_{q^{k}_{1},\dots,q^{k}_{n}}\to A$ which respects $Y$.

Recall that a matching of a graph $G$ is a subset $M\subseteq E(G)$ such that
every vertex of $G$ is incident with at most one of the edges of $M$. A
matching $M$ of $G$ is a perfect matching if every vertex of $G$ is incident
with exactly one edge in $M$. We may view a perfect matching of a graph $G$ as
a function $\mu:V(G)\to V(G)$ with the property that for any $x\in V(G)$,
$(x,\mu(x))\in E(G)$ and $\mu^{2}(x)=x$. In this sense we may speak of
continuous perfect matchings for topological graphs $G$. The following theorem
is a generalization of [7, Theorem 3.1.3] with a similar proof.

######  Theorem 3.2.

Let $S$ be a generating set of $\mathbb{Z}^{n}$. Then there is no continuous
perfect matching of $F(S,2^{\mathbb{Z}^{n}})$.

###### Proof.

Suppose $S=\\{\pm e_{1}^{\prime},\pm e_{2}^{\prime},\cdots,\pm
e_{m}^{\prime}\\}$, where none of the $e^{\prime}_{i}$ are the identity
$\bar{0}$. Rewrite $S=\\{u_{1},\dots,u_{2m}\\}$. Assume a continuous perfect
matching $\mu$ of $F(S,2^{\mathbb{Z}^{n}})$ exists. Note that the vertex set
of $F(S,2^{\mathbb{Z}^{n}})$ is the same as that of $F(2^{\mathbb{Z}^{n}})$.
Define $\eta:F(2^{\mathbb{Z}^{n}})\rightarrow S$ by letting $\eta(x)$ to be
the unique element $g\in S$ such that $\mu(x)=g\cdot x$. Then $\eta$ is
continuous.

Let $Y\subseteq S^{\mathbb{Z}^{n}}$ be the $\mathbb{Z}^{n}$-subshift of finite
type described by $2m(2m-1)$ many $\mathbb{Z}^{n}$-patterns $\\{p_{i,j}\colon
1\leq i,j\leq 2m,i\neq j\\}$, where $p_{i,j}:\\{\bar{0},u_{i}\\}\rightarrow S$
is defined by $p_{i,j}(\bar{0})=u_{i}$ and $p_{i,j}(u_{i})=-u_{j}$.

Now define

$\theta:F(2^{\mathbb{Z}^{n}})\rightarrow Y\subseteq S^{\mathbb{Z}^{n}}$

by $\theta(x)(g)=\eta(g\cdot x)$ for $x\in F(2^{\mathbb{Z}^{n}})$ and
$g\in\mathbb{Z}^{n}$. Then $\theta$ is a continuous equivariant map.

Let $q_{1},q_{2},\cdots,q_{n}$ be odd. By Theorem 3.1, for some large enough
$k$ there exists $\varphi\colon
T_{q_{1}^{k},q_{2}^{k},\cdots,q_{n}^{k}}\rightarrow S$ which respects $Y$.
Consider the $(\mathbb{Z}^{n},S)$-graph with vertex set
$T_{q_{1}^{k},q_{2}^{k},\cdots,q_{n}^{k}}$, and denote it as $T$. As $\varphi$
respects $Y$, $\varphi$ induces a perfect matching of $T$. This is a
contradiction as the cardinality of $T$ is odd. ∎

######  Corollary 3.3.

For any generating set $S$ of $\mathbb{Z}^{n}$, letting
$G=F(S,2^{\mathbb{Z}^{n}})$, then $\chi^{\prime}_{c}(G)>\chi^{\prime}(G)$.

###### Proof.

It is easy to see that
$\chi^{\prime}(G)=\chi^{\prime}(C(\mathbb{Z}^{n},S))=|S|$. Assume there is a
continuous proper edge $|S|$-coloring of $F(S,2^{\mathbb{Z}^{n}})$. Then every
vertex of $F(S,2^{\mathbb{Z}^{n}})$ has exactly one edge of each color
incident with it. Thus, for any color $u\in S$, the set of edges with color
$u$ is a continuous perfect matching of $F(S,2^{\mathbb{Z}^{n}})$,
contradicting Theorem 3.2. ∎

##  4\. Continuous Proper Edge Colorings of $F(2^{\mathbb{Z}^{n}})$

In this section we construct a continuous proper edge $(|S|+1)$-coloring of
$F(S,2^{\mathbb{Z}^{n}})$ for any $n\geq 1$ and generating set $S$ of
$\mathbb{Z}^{n}$.

Note that any generating set $S$ of $\mathbb{Z}^{n}$ has at least $2n$ many
elements. Thus throughout the rest of this section we fix once and for all
$2n+1$ many colors to be used in the construction of our edge coloring:

$c_{1},\dots,c_{n},1,\dots,n+1.$

We will name more colors as needed.

###  4.1. Proper edge colorings of a rectangle

We first work with $n$-dimensional rectangles in $\mathbb{Z}^{n}$.

We fix some more terminology about an $n$-dimensional rectangle $R$ in
$\mathbb{Z}^{n}$. We say that $R$ is of size $a_{1}\times\cdots\times a_{n}$
if there is $(b_{1},\dots,b_{n})\in\mathbb{Z}^{n}$ such that

$R=[b_{1},b_{1}+a_{1}]\times\cdots\times[b_{n},b_{n}+a_{n}].$

If $R$ is of the above form, then a vertex $x$ in $R$ is said to have
coordinates $(x_{1},\cdots,x_{n})$ in $R$ if
$x=(b_{1}+x_{1},\dots,b_{1}+x_{n})$. We view $R$ as an induced subgraph of the
Cayley graph $C(\mathbb{Z}^{n})$ of $\mathbb{Z}^{n}$ with the standard
generating set $\\{\pm e_{i}\colon 1\leq i\leq n\\}$. We say an edge $(x,y)$
in $C(\mathbb{Z}^{n})$ is parallel to $e_{i}$ if $e_{i}+x=y$ or $e_{i}+y=x$.
Let $\partial R$ denote the set of all edges in $R$ which are adjacent to at
least one edge in $C(\mathbb{Z}^{n})$ that is not in $R$. We call $\partial R$
the boundary of $R$. An edge in $C(\mathbb{Z}^{n})$ is said to be adjacent to
$R$ if it is not in $R$ but is adjacent to some edge of $R$. An edge coloring
of $C(\mathbb{Z}^{n})$ is said to satisfy the boundary condition for $R$ if

  1. (1)

it is a proper edge coloring of $R$ and its adjacent edges;

  2. (2)

for all $1\leq i\leq n$, all edges adjacent to $R$ and parallel to $e_{i}$
have the same color $c_{i}$.

$c_{1}$$c_{1}$$c_{1}$$c_{1}$$c_{1}$$c_{1}$3$c_{2}$$c_{2}$213$c_{2}$$c_{2}$121221$c_{1}$3$c_{2}$$c_{2}$
Figure 1. An edge coloring satisfying the boundary condition.

Figure 1 illustrates an example of an edge coloring of a $2$-dimensional
rectangle of size $2\times 2$ satisfying the boundary condition.

######  Lemma 4.1.

For any $n\geq 1$ and any $n$-dimensional rectangle $R$, there is a proper
edge $(2n+1)$-coloring satisfying the boundary condition for $R$.

###### Proof.

By induction on $n$. For $n=1$ the lemma is obvious. For the inductive case
$n>1$, consider an $n$-dimensional rectangle $R$ of size
$a_{1}\times\cdots\times a_{n}$. We only need to define the edge coloring for
edges either in $R$ or adjacent to $R$.

We first divide $R$ into $a_{n}+1$ many layers as illustrated in Figure 2.

...$\Big{\uparrow}$$e_{n}$ Figure 2. Layers as subgraphs of $R$.

By the inductive hypothesis, there is a proper edge $(2n-1)$-coloring of each
layer of $R$ satisfying the boundary condition. We fix such a proper edge
$(2n-1)$-coloring and use it to color all layers of $R$ identically. Note that
the colors used so far are among $c_{1},\dots,c_{n-1},1,\dots,n$, and the
remaining uncolored edges are those parallel to $e_{n}$.

Next, consider an arbitrary edge $(x,y)$ parallel to $e_{n}$. If $(x,y)$ is
adjacent to $R$, then we color it by $c_{n}$, as required by the boundary
condition. Otherwise, assume $(x,y)$ is in $R$. Then $x,y\in R$. Let $C_{x}$
be the set of colors among $c_{1},\dots,c_{n-1},1,\dots,n$ which are already
used to color an edge incident with $x$. Similarly define $C_{y}$. Then
$C_{x}=C_{y}$ and both have cardinality $2n-2$. Thus there exists a color
$c^{*}\in\\{c_{1},\dots,c_{n-1},1,\dots,n\\}$ such that $c^{*}\not\in
C_{x}=C_{y}$.

Now the uncolored edges can be decomposed into $(a_{1}+1)\cdots(a_{n-1}+1)$
many paths parallel to $e_{n}$. For each such path $p$ there is a color
$c^{*}\in\\{c_{1},\dots,c_{n-1},1,\dots,n\\}$ such that $c^{*}\not\in C_{x}$
for all vertices $x$ in $p$. We can then use the two colors $c^{*}$ and $n+1$
in alternation to color all the edges of $p$, so that the resulting edge
coloring is proper. ∎

######  Lemma 4.2.

Let $R$ be an $n$-dimensional rectangle of size $a_{1}\times\cdots\times
a_{n}$. If $a_{i}$ is odd for some $1\leq i\leq n$, then there is a proper
edge $2n$-coloring satisfying the boundary condition for $R$.

###### Proof.

We will use only the colors $c_{1},\dots,c_{n},1,\dots,n$. If $n=1$ it is easy
to see that we can use $c_{1}$ to color the two edges adjacent to $R$ and use
$c_{1}$ and $1$ in alternation to color the edges of $R$; the resulting edge
coloring is proper and satisfies the boundary condition.

Suppose $n>1$ and, without loss of generality, assume $a_{n}$ is odd.
Similarly to the proof of Lemma 4.1, we divide $R$ into $a_{n}+1$ many layers
perpendicular to $e_{n}$. By Lemma 4.1, there is a proper edge
$(2n-1)$-coloring of each layer of $R$ satisfying the boundary condition for
this layer. As in the proof of Lemma 4.1, we fix one such edge coloring and
use it to color all layers of $R$ identically. We also color any edge that is
adjacent to $R$ and parallel to $e_{n}$ by the color $c_{n}$, as required by
the boundary condition. The remaining uncolored edges are now decomposed into
$(a_{1}+1)\dots(a_{n-1}+1)$ many paths parallel to $e_{n}$. For each such path
$p$ there is a color $c^{*}\in\\{c_{1},\dots,c_{n},1,\dots,n\\}$ such that
$c^{*}\not\in C_{x}$ for all vertices $x$ in $p$, where $C_{x}$ is defined
similarly as in the proof of Lemma 4.1. Note, however, that now the path $p$
has odd length. Thus we can use the two colors $c^{*}$ and $c_{n}$ in
alternation to color all the edges of $p$, as done in the $n=1$ case. ∎

Let $R$ be an $n$-dimensional rectangle of size $a_{1}\times\cdots\times
a_{n}$, where $a_{i}$ is a positive even number for each $1\leq i\leq n$.
Suppose

$R=[b_{1},b_{1}+a_{1}]\times\cdots\times[b_{n},b_{n}+a_{n}]$

for $(b_{1},\dots,b_{n})\in\mathbb{Z}^{n}$. Let

$K=\left[b_{1}+\frac{a_{1}}{2}-1,b_{1}+\frac{a_{1}}{2}+1\right]\times\cdots\times\left[b_{n}+\frac{a_{n}}{2}-1,b_{n}+\frac{a_{n}}{2}+1\right].$

Then $K$ is a subrectangle of $R$ of size $2\times\cdots\times 2$, and we call
$K$ the core of $R$. An edge coloring of $C(\mathbb{Z}^{n})$ is said to
satisfy the core condition for $R$ if

  1. (1)

it is a proper edge $(2n+1)$-coloring of $R$ and its adjacent edges;

  2. (2)

the color $n+1$ is only used in coloring some edges of the core of $R$.

######  Lemma 4.3.

Let $k$ be a nonnegative integer and let $d=4k+2$. For any $n$-dimensional
rectangle $R$ of size $d\times\cdots\times d$, there is a proper edge
$(2n+1)$-coloring satisfying both the boundary condition and the core
condition for $R$.

###### Proof.

If $k=0$, the core of $R$ is the same as $R$ and the lemma follows immediately
from Lemma 4.1. For the rest of the proof we assume $k>0$. Again we only
define an edge coloring for edges either in $R$ or adjacent to $R$.

Our edge coloring will be defined in $n+1$ many stages. Let $R_{0}=R$. In
stage $1$, we first decompose $R_{0}$ into three parts $P_{1},Q_{1},R_{1}$ as
follows. The vertices in $P_{1}$ have their coordinates in $R_{0}$ from the
set

$[0,2k-1]\times[0,d]\times\cdots\times[0,d];$

the vertices in $R_{1}$ have their coordinates in $R_{0}$ from the set

$[2k,2k+2]\times[0,d]\times\cdots\times[0,d];$

the vertices in $Q_{1}$ have their coordinates in $R_{0}$ from the set

$[2k+3,d]\times[0,d]\times\cdots\times[0,d].$

Figure 3 is an illustration of the construction at this stage.

$\Big{\uparrow}$$e_{1}$$(2k-1)\times d\times\cdots\times d$edge
$2n$-coloring$(2k-1)\times d\times\cdots\times d$edge
$2n$-coloring$c_{1}$$c_{1}$$c_{1}$$c_{1}$$c_{1}$$c_{1}$$2\times
d\times\cdots\times d$$R_{1}$$P_{1}$$Q_{1}$ Figure 3. The first stage of the
definition of an edge coloring.

Now each of $P_{1}$ and $Q_{1}$ is a subrectangle of $R_{0}$ of size
$(2k-1)\times d\times\cdots\times d$, and $R_{1}$ is a subrectangle of $R_{0}$
of size $2\times d\times\cdots\times d$. Since $2k-1$ is odd, we can apply
Lemma 4.2 to obtain a proper edge $2n$-coloring for $P_{1}$ and for $Q_{1}$
satisfying their boundary conditions. This is the end of stage 1. Note that
the colors used so far are among $c_{1},\dots,c_{n},1,\dots,n$, and the
remaining uncolored edges are those of $R_{1}$.

In stage 2, we repeat this construction by first decomposing $R_{1}$ into
$P_{2},Q_{2},R_{2}$ according to the second coordinate of the vertices in
$R_{1}$ and then properly edge $2n$-coloring the edges either in $P_{2}$ or
$Q_{2}$ or adjacent to either of them. At the end of this stage, the remaining
uncolored edges are those of $R_{2}$ and the colors used so far are still
among $c_{1},\dots,c_{n},1,\dots,n$.

Repeating this construction $n$ times according to each of the $i$-th
coordinates, we obtain at the end of stage $n$ a subrectangle $R_{n}$ of size
$2\times\cdots\times 2$. All those edges adjacent to $R_{n}$ have been colored
to satisfy the boundary condition for $R_{n}$. Applying Lemma 4.1, we get a
proper edge $(2n+1)$-coloring of $R_{n}$ which satisfies the boundary
condition for $R_{n}$. This resulting edge coloring is proper and obviously
satisfies the core condition for $R$. ∎

Next we prove a generalization of Lemma 4.3 in which the core condition is
replaced by a shifted core condition, as follows. Let $R$ be an
$n$-dimensional rectangle of size $a_{1}\times\cdots\times a_{n}$, where
$a_{i}$ is a postive even number for each $1\leq i\leq n$. Suppose

$R=[b_{1},b_{1}+a_{1}]\times\cdots\times[b_{n},b_{n}+a_{n}]$

for $(b_{1},\dots,b_{n})\in\mathbb{Z}^{n}$. Let $K$ be the core of $R$. For
$t=(t_{1},\dots,t_{n})\in\mathbb{Z}^{n}$ where

$-\frac{a_{i}}{2}+1\leq t_{i}\leq\frac{a_{i}}{2}-1$

for each $1\leq i\leq n$, let

$K+t=\\{(x_{1}+t_{1},\dots,x_{n}+t_{n})\colon(x_{1},\dots,x_{n})\in K\\}.$

Then $K+t$ is a subrectangle of $R$ of size $2\times\cdots\times 2$. We call
$K+t$ the $t$-shifted core of $R$. An edge coloring of $C(\mathbb{Z}^{n})$ is
said to satisfy the $t$-shifted core condition for $R$ if

  1. (1)

it is a proper edge $(2n+1)$-coloring of $R$ and its adjacent edges;

  2. (2)

the color $n+1$ is only used in coloring some edges of the $t$-shifted core of
$R$.

######  Lemma 4.4.

Let $k$ be a non-negative integer and let $d=4k+2$. For any $n$-dimensional
rectangle $R$ of size $d\times\cdots\times d$ and for any
$t=(t_{1},\dots,t_{n})\in[-2k+2,2k-2]^{n}$ where $t_{i}$ is an even number for
each $1\leq i\leq n$, there is a proper edge $(2n+1)$-coloring satisfying both
the boundary condition and the $t$-shifted core condition for $R$.

###### Proof.

The proof is identical to that of Lemma 4.3, except that at each stage of the
construction $P_{i}$ will have size $2\times\cdots\times
2\times(2k-1+t_{i})\times d\times\cdots\times d$ and $Q_{i}$ will have size
$2\times\cdots\times 2\times(2k-1-t_{i})\times d\times\cdots\times d$. Since
$2k-1+t_{i}$ and $2k-1-t_{i}$ are both odd, Lemma 4.2 can be applied to
complete the proof. ∎

In our proof of the main theorem below, we view an application of Lemma 4.4 as
a two-step process. In the first step, we apply Lemma 4.3 to obtain a proper
edge $(2n+1)$-coloring of $R$ satisfying the core condition. Then, in the
second step, we may shift the core to any other position within $R$ as long as
each coordinate of the shift vector used is even. This is not quite what
happens in the proof of Lemma 4.4, but it is a useful point of view to
describe the algorithm in the proof of our main theorem.

###  4.2. Continuous proper edge colorings

In this final subsection we prove the main theorem of the paper. We will use
the following lemmas from [6].

######  Lemma 4.5.

(Basic clopen marker lemma [6, Lemma 2.1]) Let $\rho$ be the path distance on
the Schreier graph $F(2^{\mathbb{Z}^{n}})$. Then for any positive integer $d$,
there is a relatively clopen set $M_{d}\subseteq F(2^{\mathbb{Z}^{n}})$ such
that

  1. (1)

if $x,y\in M_{d}$ are distinct then $\rho(x,y)>d$;

  2. (2)

for any $x\in F(2^{\mathbb{Z}^{n}})$ there is $y\in M_{d}$ such that
$\rho(x,y)\leq d$.

The integer $d$ in the above lemma is usually called the marker distance, the
set $M_{d}$ is called a marker set, and its elements are called marker points.

######  Lemma 4.6.

(Marker regions lemma [6, Theorem 3.1]) Let $d>0$ be an integer. Then there is
a relatively clopen subequivalence relation $R^{n}_{d}$ on
$F(2^{\mathbb{Z}^{n}})$ such that each of the $R^{n}_{d}$-equivalence classes
is of the form $R\cdot x$, where $R$ is an $n$-dimensional rectangle with side
lengths either $d$ or $d+1$. Here $R^{n}_{d}$ is relatively clopen means that
$\left\\{(x,g)\in F(2^{\mathbb{Z}^{n}})\times\mathbb{Z}^{n}\colon(x,g\cdot
x)\in R^{n}_{d}\right\\}$ is a clopen subset of
$F(2^{\mathbb{Z}^{n}})\times\mathbb{Z}^{n}$.

The $R^{n}_{d}$-equivalence classes in the above lemma are called marker
regions.

All the terminology we defined in the preceding subsection about an
$n$-dimensional rectangle $R$ in $\mathbb{Z}^{n}$ can be similarly defined for
a marker region of the form $R\cdot x$ for $x\in F(2^{\mathbb{Z}^{n}})$. Thus
we may refer to a marker region as an $n$-dimensional rectangle in the
Schreier graph $F(2^{\mathbb{Z}^{n}})$. In particular, for a continuous edge
coloring of $F(2^{\mathbb{Z}^{n}})$, we may consider whether it satisfies the
boundary condition and the (shifted) core condition for a marker region, and
when we write the marker region as $R\cdot x$, the boundary condition and the
(shifted) core condition are independent from the choice of $R$ and $x$.

######  Proposition 4.7.

Let $k$ be a non-negative integer and let $d=4k+2$. Then there is a relatively
clopen subequivalence relation $R^{n}_{d}$ on $F(2^{\mathbb{Z}^{n}})$ and a
continuous proper edge $(2n+1)$-coloring of $F(2^{\mathbb{Z}^{n}})$ such that

  1. (i)

each of the $R^{n}_{d}$-equivalence classes is of the form $R\cdot x$, where
$R$ is an $n$-dimensional rectangle with side lengths either $d$ or $d+1$;

  2. (ii)

the edge coloring satisfies both the boundary condition and the core condition
for each of the $R^{n}_{d}$-equivalence classes.

###### Proof.

We apply Lemma 4.6 to obtain a relatively clopen subequivalence relation
$R^{n}_{d}$ on $F(2^{\mathbb{Z}^{n}})$ so that each of the
$R^{n}_{d}$-equivalence classes is of the form $R\cdot x$, where $R$ is an
$n$-dimensional rectangle with side lengths either $d$ or $d+1$. Consider a
fixed such $R$ and assume it is of size $a_{1}\times\cdots a_{n}$, where
$a_{i}\in\\{d,d+1\\}$ for each $1\leq i\leq n$. If any $a_{i}$ is $d+1$, we
obtain and fix a proper edge $2n$-coloring satisfying the boundary condition
for $R$ by Lemma 4.2. Otherwise, $R$ is of size $d\times\cdots\times d$, and
we obtain and fix a proper edge $(2n+1)$-coloring satisfying both the boundary
condition and the core condition for $R$ by Lemma 4.3. Using these fixed edge
colorings to color the marker regions $R\cdot x$, we obtain a continuous
proper edge $(2n+1)$-coloring of $F(2^{\mathbb{Z}^{n}})$ with both the
boundary condition and the core condition for each marker region. The
properness of the entire edge coloring follows from the boundary condition for
each marker region, and the continuity of the edge coloring follows from the
relative clopenness of $R^{n}_{d}$ and the fact that the edge coloring on each
marker region depends only on its size. ∎

In particular we obtain a continuous proper edge $(2n+1)$-coloring of
$F(2^{\mathbb{Z}^{n}})$. This generalizes [7, Theorem 3.1.4] and answers [5,
Question 10.3] for $n\geq 3$, which was also stated implicitly in [7] (see the
remark after [7, Question 3.1.5]).

######  Corollary 4.8.

For any $n\geq 1$, $\chi^{\prime}_{c}(F(2^{\mathbb{Z}^{n}}))=2n+1$.

Next, we generalize Corollary 4.8 to the case of arbitrary generating sets.
Again we assume tacitly that our generating sets are symmetric and do not
contain the identity. Let $\Lambda$ be a subgroup of $\mathbb{Z}^{n}$ and let
$S$ be a generating set of $\Lambda$, i.e., $\Lambda=\langle S\rangle$. We say
that $S$ is linearly independent if for any $s\in S$, $\langle
s\rangle\cap\langle S\setminus\\{s,-s\\}\rangle=\\{\bar{0}\\}$. If $S=2m$
where $m\leq n$ and $S$ is linearly independent, the Cayley graph
$C(\Lambda,S)$ is isomorphic to the Cayley graph $C(\mathbb{Z}^{m})$ with the
standard generating set. We have the following generalizations of Lemmas 4.5
and 4.6.

######  Lemma 4.9.

Let $\Lambda$ be a subgroup of $\mathbb{Z}^{n}$ and let $S$ be a generating
set of $\Lambda$. Assume $S$ is linearly independent. Let
$\Lambda\curvearrowright F(2^{\mathbb{Z}^{n}})$ be the restriction of the
Bernoulli shift action $\mathbb{Z}^{n}\curvearrowright F(2^{\mathbb{Z}^{n}})$
on $\Lambda\times F(2^{\mathbb{Z}^{n}})$. Let $\rho_{S}$ be the path distance
on the Schreier graph $G(\Lambda,S,F(2^{\mathbb{Z}^{n}}))$. Then for any
positive integer $d$, there is a relatively clopen set $M_{d}\subseteq
F(2^{\mathbb{Z}^{n}})$ such that

  1. (1)

if $x,y\in M_{d}$ are distinct then $\rho_{S}(x,y)>d$;

  2. (2)

for any $x\in F(2^{\mathbb{Z}^{n}})$ there is $y\in M_{d}$ such that
$\rho_{S}(x,y)\leq d$.

######  Lemma 4.10.

Let $\Lambda$ be a subgroup of $\mathbb{Z}^{n}$ and let $S$ be a generating
set of $\Lambda$. Assume $S$ is linearly independent. Let
$\Lambda\curvearrowright F(2^{\mathbb{Z}^{n}})$ be the restriction of the
Bernoulli shift action $\mathbb{Z}^{n}\curvearrowright F(2^{\mathbb{Z}^{n}})$
on $\Lambda\times F(2^{\mathbb{Z}^{n}})$. Let $d>0$ be an integer. Assume
$|S|=2m$ where $m\leq n$. Then there is a relatively clopen subequivalence
relation $R^{S}_{d}$ on $F(2^{\mathbb{Z}^{n}})$ such that each of the
$R^{S}_{d}$-equivalence classes is an $m$-dimensional rectangle in the
Schreier graph $G(\Lambda,S,F(2^{\mathbb{Z}^{n}}))$ with side lengths either
$d$ or $d+1$.

The proofs of these lemmas are identical to those of Lemmas 4.5 and 4.6. In
fact, all of our discussions about the standard generating set so far in this
paper apply to these more general linearly independent generating sets. In
particular we have the following generalization of Proposition 4.7 with the
same proof.

######  Proposition 4.11.

Let $\Lambda$ be a subgroup of $\mathbb{Z}^{n}$ and let $S$ be a generating
set of $\Lambda$. Assume $S$ is linearly independent. Let
$\Lambda\curvearrowright F(2^{\mathbb{Z}^{n}})$ be the restriction of the
Bernoulli shift action $\mathbb{Z}^{n}\curvearrowright F(2^{\mathbb{Z}^{n}})$
on $\Lambda\times F(2^{\mathbb{Z}^{n}})$. Let $k$ be a non-negative integer
and let $d=4k+2$. Assume $|S|=2m$ where $m\leq n$. Then there is a relatively
clopen subequivalence relation $R^{S}_{d}$ on $F(2^{\mathbb{Z}^{n}})$ and a
continuous proper edge $(2m+1)$-coloring of
$G(\Lambda,S,F(2^{\mathbb{Z}^{n}}))$ such that

  1. (i)

each of the $R^{S}_{d}$-equivalence classes is an $m$-dimensional rectangle
with side lengths either $d$ or $d+1$;

  2. (ii)

the edge coloring satisfies both the boundary condition and the core condition
for each of the $R^{S}_{d}$-equivalence classes.

We are now ready for the main theorem of this paper.

######  Theorem 4.12.

For any $n\geq 1$ and any generating set $S$ of $\mathbb{Z}^{n}$, let
$G=F(S,2^{\mathbb{Z}^{n}})$. Then
$\chi^{\prime}_{c}(G)=\chi^{\prime}(G)+1=|S|+1$.

###### Proof.

It is easy to see that $\chi^{\prime}(G)=|S|$. The lower bound has been
established by Corollary 3.3. We only need to construct a continuous proper
edge $(|S|+1)$-coloring of $F(S,2^{\mathbb{Z}^{n}})$.

Write $S=\bigcup_{i=0}^{m}S_{i}$ so that each $S_{i}$ is a maximal linearly
independent subset of $S\setminus\bigcup_{j=0}^{i-1}S_{j}$. Then $m\geq 0$,
and for each $0\leq i\leq m$, $S_{i}$ is symmetric. In particular,
$|S_{0}|=2n$. For each $1\leq i\leq m$, let $|S_{i}|=2n_{i}$ and let $k_{i}>0$
be the smallest natural number such that $k_{i}\langle
S_{i}\rangle\subseteq\langle S_{i-1}\rangle$.

Let $\alpha=3^{n}m$ and $\beta=2\cdot 3\cdot\prod_{i=1}^{m}k_{i}$. Fix an
arbitrary non-identity $s=(s_{1},\dots,s_{n})\in S_{m}$ and let
$|s|=\sum_{k=1}^{n}|s_{k}|$. For each $1\leq i\leq m$, since $\beta$ is a
multiple of $\prod_{j=i+1}^{m}k_{j}$, we have $\beta
s\in\bigcap_{j=i}^{m}\langle S_{j}\rangle$; in particular $\beta s\in\langle
S_{i}\rangle$. Thus there exist $a_{i,1},\dots,a_{i,n_{i}}\in\mathbb{Z}$ such
that $\beta s=a_{i,1}e_{i,1}+\cdots+a_{i,n_{i}}e_{i,n_{i}}$, where
$S_{i}=\\{\pm e_{i,1},\dots,\pm e_{i,n_{i}}\\}$. Let
$\gamma=\sup\\{|a_{i,j}|\colon 1\leq i\leq m,1\leq j\leq n_{i}\\}$. Let
$d=4\cdot 2(\gamma+1)(\alpha+1)(\beta+1)|s|+2$.

Consider the Schreier graph $G_{i}=G(\langle
S_{i}\rangle,S_{i},F(2^{\mathbb{Z}^{n}}))$. Let $\rho_{i}=\rho_{S_{i}}$ be the
path distance on $G_{i}$. We have $E(G)=\bigcup_{i=0}^{m}E(G_{i})$. For each
$0\leq i\leq m$, let $H_{i}$ be the Schreier graph on $F(2^{\mathbb{Z}^{n}})$
where $E(H_{i})=\bigcup_{j=i}^{m}E(G_{j})$, and let $C_{i}$ be a set of
$2n_{i}$ many fresh colors, i.e., for $i\neq j$, $C_{i}\cap
C_{j}=\varnothing$. Assume $0$ is a color distinct from any color in
$\bigcup_{i=0}^{m}C_{i}$. Let $N_{i}=1+\sum_{j=i}^{m}2n_{j}$. To define a
continuous edge $(|S|+1)$-coloring on $G$, we use reverse induction on
$i=m,\dots,0$ to define a continuous edge $N_{i}$-coloring of $H_{i}$ with the
colors in $\\{0\\}\cup\bigcup_{j=i}^{m}C_{j}$. When $i=0$ the resulting edge
coloring is as required.

When $i=m$ we apply Proposition 4.11 to $S_{m}$ to obtain a relatively clopen
subequivalence relation $R^{m}_{d}=R^{S_{m}}_{d}$ on $F(2^{\mathbb{Z}^{n}})$
and a continuous proper edge $N_{m}$-coloring of $H_{m}=G_{m}$. Let
$\kappa_{m}$ denote this coloring. Then $\kappa_{m}$ uses colors in
$\\{0\\}\cup C_{m}$. For each $R^{m}_{d}$-marker region, which is an
$n_{m}$-dimensional rectangle with side lengths either $d$ or $d+1$,
$\kappa_{m}$ satisfies both the boundary condition and the core condition. In
particular, the boundary condition guarantees that $\kappa_{m}$ is a proper
edge coloring of $G_{m}$. Here the core condition means that the color $0$ is
only used in coloring some edges of the cores of $R^{m}_{d}$-marker regions.
Let $K_{m}$ be the set of all cores of the $R^{m}_{d}$-marker regions. Then
$K_{m}$ is a clopen subset of $F(2^{\mathbb{Z}^{n}})$.

For the clarity of our proof, consider $i=m-1$ next. Now we apply Proposition
4.11 to $S_{m-1}$ to obtain a relatively clopen subequivalence relation
$R^{m-1}_{d}=R^{S_{m-1}}_{d}$ on $F(2^{\mathbb{Z}^{n}})$ and a continuous
proper edge $(2n_{m-1}+1)$-coloring of $G_{m-1}$. Let $\eta_{m-1}$ denote this
coloring. Then $\eta_{m-1}$ uses colors in $\\{0\\}\cup C_{m-1}$. For each
$R^{m-1}_{d}$-marker region, which is an $n_{m-1}$-dimensional rectangle with
side lengths $d$ or $d+1$, $\eta_{m-1}$ satisfies both the boundary condition
and the core condition. Here again the core condition means that the color $0$
is only used in coloring some edges of the cores of $R^{m-1}_{d}$-marker
regions. Let $Q_{m-1}$ be the set of all cores of the $R^{m-1}_{d}$-marker
regions. Then $Q_{m-1}$ is a clopen subset of $F(2^{\mathbb{Z}^{n}})$.

Consider a particular $R^{m-1}_{d}$-marker region $R$ and its core $K$. We
claim that one of the following $\alpha+1$ many sets

$K,\ \beta s\cdot K,\ 2\beta s\cdot K,\ \dots,\ \alpha\beta s\cdot K$

has empty intersection with $K_{m}$. To see this, note that $\beta$ is a
multiple of $k_{m}$, and hence $\beta s\in\langle S_{m-1}\rangle$. Since $K$
is an $n_{m-1}$-dimensional rectangle in $G_{m-1}$ with side lengths $2$, and
$\beta$ is a multiple of $3$, we have that the above $\alpha+1$ many sets are
pairwise disjoint. Since $d>4(\alpha+1)\gamma\beta|s|$, all of the above
$\alpha+1$ many sets are still contained in $R$. On the other hand, for each
point $x\in K$, consider

$x,\ \beta s\cdot x,\ 2\beta s\cdot x,\ \dots,\ \alpha\beta s\cdot x.$

If any of these $\alpha+1$ many points is in $K_{m}$, say in the core of a
particular $R^{m}_{d}$-marker region $R^{\prime}$, then all of the displayed
$\alpha+1$ many points are still in $R^{\prime}$, and none of the other points
are in $K_{m}$ anymore. This means that at most one of the displayed
$\alpha+1$ many points can be in $K_{m}$. Now, since $|K|=3^{n_{m-1}}\leq
3^{n}\leq\alpha$, there are at most $\alpha$ many $R^{m}_{d}$-marker regions
with which any of the displayed $\alpha+1$ many shifts of $K$ has a nonempty
intersection. The claim follows from the pigeonhole principle.

Now we define the coloring $\kappa_{m-1}$, which is an extension of
$\kappa_{m}$. For each $R^{m-1}_{d}$-marker region $R$ and its core
$K\subseteq Q_{m-1}$, we find the least integer $a$ such that $0\leq
a\leq\alpha$ and $a\beta s\cdot K\cap K_{m}=\varnothing$. If $R$ has an side
of length $d+1$ then we let $\kappa_{m-1}$ on $R$ and its adjacent edges to be
$\eta_{m-1}$ on $R$ and its adjacent edges. Note that only colors in $C_{m-1}$
are used in $\eta_{m-1}$. Otherwise, apply Lemma 4.4 to obtain a proper edge
$(2n_{m-1}+1)$-coloring $\kappa_{m-1}$ on $R$ satisfying both the boundary
condition and the $a\beta s$-shifted core condition for $R$. Again the shifted
core condition means that the color $0$ is only used in coloring some edges of
$a\beta s\cdot K$. This is possible since $d=4k+2$ for some $k$ and $a\beta$
is even. The shifted core condition guarantees that no adjacent edges are both
colored $0$ in the resulting $\kappa_{m-1}$. Furthermore, no adjacent edges
are colored the same color in $\kappa_{m-1}$ since $C_{m-1}\cap
C_{m}=\varnothing$. Together with this observation, the boundary condition in
the definition guarantees that $\kappa_{m-1}$ is a proper edge
$N_{m-1}$-coloring of $H_{m-1}$. It is clear from the definition that
$\kappa_{m-1}$ is continuous. This finishes the definition of $\kappa_{i}$ as
a continuous proper edge $N_{i}$-coloring of $H_{i}$ for $i=m-1$. To set up
the general induction, we let $K_{m-1}$ be the set of all shifted cores of the
$R^{m-1}_{d}$-marker regions as defined above. Then $K_{m-1}$ is still a
clopen subset of $F(2^{\mathbb{Z}^{n}})$. Note that the original cores in
$Q_{m-1}$ can be shifted to at most $\gamma\alpha\beta|s|$ away to form
$K_{m-1}$; however, since $d>4\cdot 2\gamma\alpha\beta|s|$, each shifted core
in $K_{m-1}$ is still at least $\gamma\alpha\beta|s|$ away from the boundaries
of the $R^{m-1}_{d}$-marker regions.

For the general $i<m-1$, assume inductively that we have defined a continuous
proper edge $N_{i+1}$-coloring $\kappa_{i+1}$ of $H_{i+1}$. Also assume that
for all $i+1\leq j\leq m$ we have defined relatively clopen subequivalence
relations $R^{j}_{d}$ of $F(2^{\mathbb{Z}^{n}})$ and relatively clopen sets
$K_{j}$ as shifted cores of $R^{j}_{d}$-marker regions.

Apply Proposition 4.11 to $S_{i}$ to obtain a relatively clopen subequivalence
relation $R^{i}_{d}=R^{S_{i}}_{d}$ on $F(2^{\mathbb{Z}^{n}})$ and a continuous
proper edge $(2n_{i}+1)$-coloring of $G_{i}$. Let $\eta_{i}$ denote this
coloring. Then $\eta_{i}$ uses colors in $\\{0\\}\cup C_{i}$. For each
$R^{i}_{d}$-marker region, which is an $n_{i}$-dimensional rectangle with side
lengths $d$ or $d+1$, $\eta_{i}$ satisfies both the boundary condition and the
core condition. For a particular $R^{i}_{d}$-marker region $R$ and its core
$K$, we claim that at least one of the $\alpha+1$ many sets

$K,\ \beta s\cdot K,\ 2\beta s\cdot K,\ \dots,\ \alpha\beta s\cdot K$

has empty intersection with $\bigcup_{j=i+1}^{m}K_{j}$. The argument for this
claim is similar to the above proof, except that we now use the fact $\beta
s\in\langle S_{i}\rangle$ to see that the above $\alpha+1$ many displayed sets
are still in $R$. For any $x\in K$ consider the points

$x,\ \beta s\cdot x,\ 2\beta s\cdot x,\ \dots,\ \alpha\beta s\cdot x.$

For any $j$ with $i+1\leq j\leq m$, if any of these points is in $K_{j}$,
which is in some $R^{j}_{d}$-marker region $R^{\prime}$, then by induction,
the entire sequence is still at least $\gamma\alpha\beta|s|$ away from the
boundary of $R^{\prime}$, and therefore still contained in $R^{\prime}$. It
follows that at most one of these points can be in $K_{j}$, and thus at most
$m$ many of these points can be in $\bigcup_{j=i+1}^{m}K_{j}$. Now since
$|K|=3^{n_{i}}\leq 3^{n}$, there are at most $3^{n}m=\alpha$ many points in
the displayed $\alpha+1$ many shifts of $K$ which can be in
$\bigcup_{j=i+1}^{m}K_{j}$. It follows that there are at most $\alpha$ many of
the displayed sets which can have nonempty intersection with
$\bigcup_{j=i+1}^{m}K_{j}$. The claim again follows from the pigeonhole
principle.

We define the coloring $\kappa_{i}$ as an extension of $\kappa_{i+1}$. For
each $R^{i}_{d}$-marker region $R$ and its core $K$, we find the least integer
$a$ such that $0\leq a\leq\alpha$ and $a\beta s\cdot
K\cap\bigcup_{j=i+1}^{m}K_{j}=\varnothing$. If $R$ has an side of length $d+1$
then we let $\kappa_{i}$ on $R$ and its adjacent edges to be $\eta_{i}$. Note
that in this case only colors in $C_{i}$ are used. Otherwise, apply Lemma 4.4
to obtain a proper edge $(2n_{i}+1)$-coloring $\kappa_{i}$ on $R$ satisfying
both the boundary condition and the $a\beta s$-shifted core condition for $R$.
This is possible since $d=4k+2$ for some $k$ and $a\beta$ is even. The shifted
core condition guarantees that no adjacent edges are both colored $0$ in the
resulting $\kappa_{i}$. Furthermore, no adjacent edges are colored the same
color in $\kappa_{i}$ since $C_{i}\cap C_{j}=\varnothing$ for all $i+1\leq
j\leq m$. Together with this observation, the boundary condition in the
definition guarantees that $\kappa_{i}$ is a proper edge $N_{i}$-coloring of
$H_{i}$. It is clear from the definition that $\kappa_{i}$ is continuous. This
finishes the definition of $\kappa_{i}$ as a continuous proper edge
$N_{i}$-coloring of $H_{i}$. To continue the induction, we let $K_{i}$ be the
set of all shifted cores of the $R^{i}_{d}$-marker regions as defined above.
Then $K_{i}$ is still a clopen subset of $F(2^{\mathbb{Z}^{n}})$.

To complete the proof, note that $\kappa_{0}$ is a continuous proper edge
$N_{0}$-coloring of $H_{0}$, where $H_{0}=F(S,2^{\mathbb{Z}^{n}})$ and
$N_{0}=|S|+1$. ∎

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# The Hi mass function of star-forming galaxies at $\mathbf{z\sim 0.35}$

Apurba Bera  Inter-University Centre for Astronomy and Astrophysics, Pune
411007, India  National Centre for Radio Astrophysics, Tata Institute of
Fundamental Research, Pune 411007, India  Nissim Kanekar  National Centre for
Radio Astrophysics, Tata Institute of Fundamental Research, Pune 411007, India
Jayaram N. Chengalur  National Centre for Radio Astrophysics, Tata Institute
of Fundamental Research, Pune 411007, India  Jasjeet S. Bagla  Indian
Institute of Science Education and Research Mohali, Knowledge City, Sector 81,
Sahibzada Ajit Singh Nagar, Punjab 140306, India

###### Abstract

The neutral atomic hydrogen (Hi) mass function (HiMF) describes the
distribution of the Hi content of galaxies at any epoch; its evolution
provides an important probe of models of galaxy formation and evolution. Here,
we report Giant Metrewave Radio Telescope Hi 21cm spectroscopy of blue star-
forming galaxies at $z\approx 0.20-0.42$ in the Extended Groth Strip, which
has allowed us to determine the scaling relation between the average Hi mass
($\rm{M_{H{\textsc{i}}}}$) and the absolute B-band magnitude ($\rm{M_{B}}$) of
such galaxies at $z\approx 0.35$, by stacking the Hi 21cm emission signals of
galaxy subsamples in different $\rm{M_{B}}$ ranges. We combine this
$\rm{M_{H{\textsc{i}}}-M_{B}}$ scaling relation (with a scatter assumed to be
equal to that in the local Universe) with the known B-band luminosity function
of star-forming galaxies at these redshifts to determine the HiMF at $z\approx
0.35$. We show that the use of the correct scatter in the
$\rm{M_{H{\textsc{i}}}-M_{B}}$ scaling relation is critical for an accurate
estimate of the HiMF. We find that the HiMF has evolved significantly from
$z\approx 0.35$ to $z\approx 0$, i.e. over the last four Gyr, especially at
the high-mass end. High-mass galaxies, with $\rm{M_{H{\textsc{i}}}\gtrsim
10^{10}\ M_{\odot}}$, are a factor of $\approx 3.4$ less prevalent at
$z\approx 0.35$ than at $z\approx 0$. Conversely, there are more low-mass
galaxies, with $\rm{M_{H{\textsc{i}}}\approx 10^{9}\ {M}_{\odot}}$, at
$z\approx 0.35$ than in the local Universe. While our results may be affected
by cosmic variance, we find that massive star-forming galaxies have acquired a
significant amount of Hi through merger events or accretion from the
circumgalactic medium over the past four Gyr.

Galaxy evolution — Radio spectroscopy — Neutral atomic hydrogen

##  1 Introduction

The neutral atomic hydrogen (Hi) mass function (HiMF) of galaxies, the number
of galaxies with a given Hi mass per unit cosmic volume, gives the
distribution of Hi across galaxies in the Universe and is a fundamental
descriptor of galaxy populations. Observational constraints on the redshift
evolution of the HiMF are essential to test the predictions of different
galaxy evolution models (e.g. Somerville & Primack, 1999; Cole et al., 2000;
Bahé et al., 2016; Davé et al., 2017, 2020). This is especially the case in
the modern view of galaxies in which the baryon cycle, the exchange of gas
between a galaxy and its circumgalactic medium (CGM), plays a critical role in
galaxy evolution (e.g. Péroux & Howk, 2020). However, the HiMF has so far only
been determined at the present epoch, $z\approx 0$ (e.g. Zwaan et al., 2005;
Jones et al., 2018); the lack of observational constraints beyond the local
Universe has meant that its redshift evolution is entirely unknown. A direct
measurement of the HiMF at cosmological distances requires the detection of Hi
21 cm emission from a large number of individual galaxies at the redshift of
interest. Unfortunately, this is very difficult to accomplish with present-day
radio telescopes, due to the low Einstein-A coefficient of the hyperfine Hi 21
cm transition. Indeed, only a single galaxy has so far been detected in Hi 21
cm emission at $z\gtrsim 0.3$ (e.g. Fernández et al., 2016).

Over the last few decades, several studies have found that the Hi mass ($\rm
M_{H{\textsc{i}}}$) of galaxies in the local Universe correlates with their
optical luminosity (e.g. Haynes & Giovanelli, 1984; Toribio et al., 2011); the
tightest correlation is between $\rm M_{H{\textsc{i}}}$ and the absolute
B-band magnitude, $\rm M_{B}$ (Dénes et al., 2014). Briggs (1990) pointed out
that such a scaling relation between Hi mass and $\rm M_{B}$ could be combined
with the B-band luminosity function of galaxies to determine the HiMF. Indeed,
this was the approach originally used to determine the HiMF at $z\approx 0$,
before the era of unbiased wide-field Hi 21 cm emission surveys (Rao & Briggs,
1993; Zwaan et al., 2001). Deep optical photometry has yielded accurate B-band
luminosity functions for large samples of galaxies out to high redshifts,
$z\gtrsim 1$ (e.g. Willmer et al., 2006; López-Sanjuan et al., 2017). A
measurement of the $\rm M_{H{\textsc{i}}}-\rm M_{B}$ relation in galaxies at
cosmological distances might then be combined with the B-band luminosity
function to measure the HiMF at these epochs.

Although it is difficult to detect Hi 21 cm emission from individual galaxies
at $z\gtrsim 0.2$ with current radio telescopes, the method of Hi 21 cm
emission “stacking” allows us to measure the average Hi mass and average Hi
properties of galaxy samples at cosmological distances (e.g. Zwaan, 2000;
Chengalur et al., 2001; Lah et al., 2007; Delhaize et al., 2013; Rhee et al.,
2016, 2018; Kanekar et al., 2016; Bera et al., 2019; Chowdhury et al., 2020,
2021, 2022a, 2022b, 2022c; Sinigaglia et al., 2022). Such Hi 21 cm stacking
experiments also allow one to determine galaxy scaling relations, the average
Tully-Fisher relation, etc (e.g. Fabello et al., 2011, 2012; Meyer et al.,
2016).

We have used the Giant Metrewave Radio Telescope to carry out a deep, $\approx
350$-hr, Hi 21 cm emission survey of the Extended Groth Strip (EGS; Bera et
al., 2019, Bera et al., in prep.). In this Letter, we present the first
measurement of the $\rm M_{H{\textsc{i}}}-\rm M_{B}$ relation in star-forming
galaxies at cosmological distances, $z\approx 0.35$. We further use this
relation along with the B-band luminosity function of galaxies at $z\approx
0.2-0.4$ to obtain the first determination of the HiMF at cosmological
distances. Throughout this work, we use a flat $\Lambda$-cold dark matter
($\Lambda$CDM) cosmology, with ($\rm H_{0}$, $\rm\Omega_{m}$,
$\rm\Omega_{\Lambda})=(70$ km s-1 Mpc-1, $0.3,0.7)$. All magnitudes are in the
AB system (Oke, 1974).

##  2 Observations and data analysis

We used the Band-5 receivers of the upgraded GMRT (Gupta et al., 2017) to
observe the EGS over four observing cycles between March 2017 and June 2019,
in proposals 31_038 (P.I. J. S. Bagla), 34_083 (P.I. N. Kanekar), 35_085 (P.I.
A. Bera), and 36_064 (P.I. A. Bera). The total observing time was $\approx
347$ hours, with a total on-source time of $\approx 250$ hours. The
observations and data analysis are described in detail by Bera et al. (in
prep.); a brief summary is provided below.

Our GMRT observations of the EGS covered the frequency range $970-1370$ MHz,
using the GMRT Wideband Backend (GWB) as the correlator, with a bandwidth of
400 MHz sub-divided into 8,192 spectral channels. The basic data editing, to
remove non-working antennas and data affected by systematic effects such as
radio frequency interference (RFI), and calibration to determine the antenna-
based gains and bandpasses were carried out in the classic aips package
(Greisen, 2003). For each observing cycle, the calibrated EGS visibilities
were then combined into a single data file. A self-calibration and data
editing procedure was run on each data set, involving a few rounds of imaging
and phase self-calibration, followed by a couple of rounds of imaging and
amplitude-and-phase self-calibration, and data editing. This process was
continued until both the antenna-based gains and the residual visibilities did
not improve on further self-calibration and data editing. Following this, all
detected continuum emission was subtracted from the calibrated spectral-line
visibilities, using the aips task uvsub. The residual visibilities were then
imaged using the task tclean in the casa package (version 5.6; McMullin et
al., 2007) to produce spectral-line data cubes in the barycentric frame; this
imaging was carried out using w-projection (Cornwell et al., 2008) and Briggs
weighting, with robust=0.5 (Briggs, 1995). Each observing cycle yielded one
independent spectral-line data cube. A frequency-dependent correction for the
shape of the antenna primary beam was then applied to each cube. The final
spectral cubes have a frequency resolution of 97.7 kHz, corresponding to a
velocity resolution of $\approx 21-30$ km s-1 across the observing band. The
synthesized beams of the cubes have FWHMs of $\approx 2\farcs 6-3\farcs 3$,
corresponding to spatial resolutions of $\approx 9-18$ kpc for the redshift
range $z=0.20-0.42$.

##  3 Hi 21 cm spectral line stacking

###  3.1 Sample selection and the Hi 21 cm subcubes

The EGS was chosen as the target of our Hi 21 cm emission survey due to the
availability of accurate spectroscopic redshifts for a large number of
galaxies with ${\rm R_{AB}}\leq 24.1$ at $z\approx 0.20-0.42$, from the DEEP2
and DEEP3 Galaxy Surveys (Cooper et al., 2012; Newman et al., 2013). The
redshift accuracy is sufficient (velocity errors $\lesssim 62$ km s-1; Newman
et al., 2013) to allow stacking of the Hi 21 cm emission signals of the
individual galaxies, without dilution of the stacked Hi 21 cm signal (e.g.
Elson et al., 2019). The DEEP2 and DEEP3 redshift coverage implies the limit
$z\geq 0.2$ on our sample, while the frequency coverage of the GMRT Band-5
receivers gives the limit $z\lesssim 0.42$; our stacking analysis was hence
restricted to galaxies with $0.20\leq z\leq 0.42$. We further restricted our
sample to galaxies (1) lying within the half-power point of the GMRT primary
beam at each galaxy’s redshifted Hi 21 cm line frequency, (2) with reliable
redshifts, with redshift quality code, Q $\geq 3$ (Newman et al., 2013), and
(3) with absolute blue magnitude $\rm M_{B}\leq-16$. This yielded a sample of
808 galaxies. The presence of an active galactic nucleus (AGN) in a galaxy
could affect its average Hi properties; we hence aimed to exclude galaxies
containing AGNs from the sample. AGNs have been shown to dominate the
population of radio sources with rest-frame 1.4 GHz radio luminosity $L_{\rm
1.4\;GHz}\gtrsim 2\times 10^{23}$ W Hz-1 (Condon et al., 2002; Smolčić et al.,
2008). We applied this radio luminosity threshold to our continuum image of
the EGS to identify, and then exclude, possible AGNs from our sample. In
addition, galaxies identified as AGN hosts in the DEEP2 or DEEP3 catalogues,
based on their optical spectra, were also excluded from the sample. A total of
84 AGNs were identified and excluded by these criteria. Next, using the DEEP2
colour criterion ${\rm U-B}+0.032\times({\rm M_{B}}+21.62)-1.035<0$ (where U
and B are the absolute magnitudes in the rest-frame U and B bands; Willmer et
al., 2006; Coil et al., 2008), we excluded 101 red galaxies, restricting our
sample to blue star-forming galaxies at $0.20\leq z\leq 0.42$.

For each target galaxy in our sample, we extracted a subcube centred at the
position and redshifted Hi 21 cm line frequency of the galaxy from each of the
final four spectral cubes of the four observing cycles. The four subcubes of
each target galaxy were treated independently in the analysis, so that any
systematic errors in one of the observing cycles would not cause all data on
the galaxy to be dropped from our sample (e.g. Chowdhury et al., 2022b; Bera
et al., in prep.). The subcubes were converted from flux density (in Jy) to
luminosity density (in Jy Mpc2), using the relation $L_{{\rm
H}{\textsc{i}}}={4\pi.F_{\nu}.d_{L}^{2}(z)}/(1+z)$, where $d_{L}(z)$ is the
luminosity distance and $F_{\nu}$ is the flux density.

Each subcube was convolved with a Gaussian beam to a spatial resolution of 30
kpc at the galaxy redshift, and then appropriately normalized to ensure that
the peak of the convolved (with the same kernel) point spread function is
unity. The spatial resolution of 30 kpc was chosen to ensure that the stacked
Hi 21 cm emission signal is not resolved: smoothing the subcubes to coarser
resolutions yielded a total Hi 21 cm emission signal consistent with that
obtained at a resolution of 30 kpc (Bera et al., in prep.). Each subcube was
also interpolated to a velocity resolution of 30 km s-1, and regridded to a
spatial pixel size of 5 kpc at the galaxy redshift. A second-order polynomial
was then fitted to the spectrum at each spatial pixel of each subcube,
excluding the central $\pm 200$ km s-1, and subtracted out. The residual
subcubes have spatial extents of $500\ {\rm kpc}\times 500\ {\rm kpc}$, and
cover the velocity range $\pm 1000$ km s-1, at the galaxy redshift.

Finally, a set of statistical tests were used to test the subcubes for the
presence of systematic effects arising due to, e.g., RFI, deconvolution
errors, etc (see Bera et al., in prep., for details). Subcubes that were
identified as being affected by such systematics were excluded from the
sample. Additionally, to mitigate the possible effect of source confusion, 14
galaxies with ‘neighbours’ were excluded from the sample. A ‘neighbour’ is
defined as any galaxy with $\rm M_{B}\leq-16$ within a spatial resolution
element ($30$ kpc) of the target galaxy and within $\pm$ 300 km s-1 in
redshift. This yielded a final sample of 464 unique galaxies, with 1665
subcubes, for our Hi 21 cm stacking experiment. The average redshift of the
sample is $\langle z\rangle=0.35$.

###  3.2 Stacking the Hi 21 cm emission

To determine the $\rm M_{H{\textsc{i}}}-M_{B}$ relation at $z\sim 0.35$, we
first divided the galaxy sample into three independent $\rm M_{B}$ bins. The
number of $\rm M_{B}$ bins and the bin widths were selected to ensure that the
average Hi 21 cm signal is detected at $>4\sigma$ significance in each
bin111We note that slight changes in the ranges of the three $\rm M_{B}$ bins
were found to have no significant effect on the inferred $\rm
M_{H{\textsc{i}}}-\rm M_{B}$ scaling relation.. The $\rm M_{B}$ ranges of the
three bins are listed in the first column of Table 1.

The stacking of the Hi 21 cm emission signals was carried out independently
for each $\rm M_{B}$ bin. For each bin, the subcubes of all galaxies in the
bin were stacked, plane-by-plane, to produce a stacked spectral cube. The
stacking was done without assigning any weights to the different subcubes,
i.e. all of them were given equal weights. This was done because weighting
based on the RMS noise in flux density units would give high weights to the
relatively few galaxies in the central regions of the field. Conversely,
weighting based on the RMS noise in luminosity density units would give high
weights to galaxies at the lowest redshifts. Further, the errors on the final
HI masses are dominated by the jackknife errors (see Table 1), rather than by
the errors on the stacked Hi 21 cm emission signals. Given that the lack of
weights simplifies the interpretation of the stacked emission signal, we chose
to not use weights in the stacking process. A second-order polynomial was then
fitted to the spectrum at each spatial pixel of the cubes, excluding the
central $\pm 200$ km s-1, and subtracted out. Finally, each residual cube was
Hanning-smoothed to, and resampled at, a velocity resolution of 60 km s-1, to
obtain the final stacked Hi 21 cm spectral cube for each bin.

The stacked Hi 21 cm spectrum for each $\rm M_{B}$ bin was obtained by taking
cuts through the central pixel of the three final stacked cubes. The final
stacked spectra for the three $\rm M_{B}$ bins are shown in Figure 1. In all
cases, the stacked Hi 21 cm emission signal is detected at $>4\sigma$
significance. For each bin, the average Hi 21 cm line luminosity was
determined by integrating the stacked Hi 21 cm spectrum over all contiguous
central spectral channels with $\geq 1.5\sigma$ significance. The RMS noise in
the planes of the corresponding stacked cube were used as the measurement
errors to estimate the detection significance of the average Hi 21 cm signal
for each bin. The velocity-integrated stacked Hi 21 cm line luminosities are
listed in the penultimate column of Table 1.

Finally, the stacked Hi 21 cm line luminosities were converted to the average
Hi mass of the galaxies in each $\rm M_{B}$ bin via the relation $\rm
M_{H{\textsc{i}}}=1.86\times 10^{4}\;\int L_{{\rm H}{\textsc{i}}}\>dv$, where
$\rm M_{H{\textsc{i}}}$ is in $\rm M_{\odot}$ and $\rm\int L_{{\rm
H}{\textsc{i}}}\>dv$ is in Jy Mpc2 km s-1. Jackknife re-sampling was used to
estimate the uncertainties on the average Hi masses; these are larger than the
measurement errors, and include contributions from both sample variance and
any underlying systematic effects. The final average Hi masses and errors are
listed in the last column of Table 1.

We also stacked the Hi 21 cm emission from all 464 galaxies of our sample
(Bera et al., in prep.). The mean Hi 21 cm line luminosity of the full sample
is $(8.4\pm 1.2)\times 10^{4}$ Jy Mpc2 km s-1 (measurement errors), while the
corresponding mean Hi mass is $(1.57\pm 0.26)\times 10^{9}\;\rm M_{\odot}$
(jackknife errors). We also measured the median Hi mass of the full sample
using a median-stacking approach222Median-stacking refers to taking the median
(instead of the mean) of the corresponding pixels, plane-by-plane, of the
individual subcubes.. This yielded a median Hi 21 cm line luminosity of
$(6.3\pm 1.0)\times 10^{4}$ Jy Mpc2 km s-1 (measurement errors); the
corresponding median Hi mass is $(1.17\pm 0.22)\times 10^{9}\>\rm M_{\odot}$
(jackknife errors).

Figure 1: The stacked Hi 21 cm emission spectra (in luminosity density) for
blue galaxies in the three $\rm M_{B}$ bins, [A] $-16.0\geq\rm M_{B}>-17.6$,
[B] $-17.6\geq\rm M_{B}>-19.2$, and [C] $-19.2\geq\rm M_{B}\geq-22.0$. The
dashed black curves in each panel show the RMS noise in the corresponding
velocity planes of the stacked spectral cubes. See text for details. Table 1:
The average Hi mass of blue galaxies in different $\rm M_{B}$ bins.

$\rm M_{B}$ range | Number of galaxies | Median $\rm M_{B}$ | $\int L_{{\rm H}{\textsc{i}}}\>dv$ | $\langle\rm M_{H{\textsc{i}}}\rangle$
---|---|---|---|---
|  |  | $10^{5}\>{\rm Jy\ Mpc^{2}\ km\ s^{-1}}$ | $10^{9}\>{\rm M}_{\odot}$
$(-17.6,-16.0]$ | 190 | $-17.03$ | $0.59\pm 0.14$ | $1.09\pm 0.29$
$(-19.2,-17.6]$ | 195 | $-18.24$ | $0.85\pm 0.19$ | $1.59\pm 0.43$
$[-22.0,-19.2]$ | 79 | $-19.86$ | $1.91\pm 0.35$ | $3.56\pm 0.81$

The errors on the velocity-integrated Hi 21 cm line luminosities in the
penultimate column are measurement errors. The errors on the Hi mass in the
last column are based on jackknife resampling.

##  4 The $\rm M_{H{\textsc{i}}}-\rm M_{B}$ scaling relation at $z\approx
0.35$

Figure 2: The $\mathbf{M_{HI}-M_{B}}$ relation for blue galaxies at
$\mathbf{z\sim 0.35}$: The solid blue circles show our estimates of
$\rm\langle log(\rm M_{H{\textsc{i}}})\rangle$ values in blue galaxies in the
three $\rm M_{B}$ bins at $z\approx 0.35$ plotted against the median $\rm
M_{B}$ values. The error bars are estimated by jackknife re-sampling. The
horizontal error bars show the bin widths. The solid black line shows the
best-fit $\rm M_{H{\textsc{i}}}-\rm M_{B}$ scaling relation at $z\approx
0.35$, while the blue shaded region shows the 68% confidence interval around
the scaling relation. The red dashed line indicates the $\rm
M_{H{\textsc{i}}}-\rm M_{B}$ relation at $z\approx 0$ (Dénes et al., 2014).

For galaxies in the local Universe, the dependence of the Hi mass with the
absolute B-band luminosity is well described by a linear relation between the
logarithm of the Hi mass, $\rm log[\rm M_{H{\textsc{i}}}]$, and the absolute
B-band magnitude (e.g. Dénes et al., 2014). The best-fit scaling relation can
be directly obtained from the measurements of the Hi masses of individual
galaxies. The local $\rm M_{H{\textsc{i}}}-\rm M_{B}$ scaling relation (for
late-type galaxies) is well described by the parametric form

${\rm log}\left(\frac{\rm M_{H{\textsc{i}}}}{{\rm
M}_{\odot}}\right)=\alpha+\beta\times\rm M_{B}$ (1)

where $\beta=-0.34\pm 0.01$ and $\alpha=2.89\pm 0.11$ (Dénes et al., 2014).
The scaling relation has a logarithmic scatter of $\sigma=0.26$ dex (Dénes et
al., 2014).

The $\rm M_{H{\textsc{i}}}-\rm M_{B}$ scaling relation of Equation 1 yields
the average value of $\rm log[\rm M_{H{\textsc{i}}}]$, i.e. $\langle\rm
log[\rm M_{H{\textsc{i}}}]\rangle$, at different values of $\rm M_{B}$.
However, in the present analysis, the method of Hi 21 cm stacking yields the
average value of $\rm M_{H{\textsc{i}}}$, i.e. $\langle\rm
M_{H{\textsc{i}}}\rangle$, for each $\rm M_{B}$ bin, and we can thus directly
obtain $\rm log[\langle\rm M_{H{\textsc{i}}}\rangle]$, rather than $\langle\rm
log[\rm M_{H{\textsc{i}}}]\rangle$. The relation between $\rm log[\langle\rm
M_{H{\textsc{i}}}\rangle]$ and $\langle\rm log[\rm M_{H{\textsc{i}}}]\rangle$
in an $\rm M_{B}$ bin depends on the distribution of the Hi masses of the
galaxies in that bin. Assuming that the Hi masses of the galaxies within a bin
are log-normally distributed, we have

$\langle\rm log[\rm M_{H{\textsc{i}}}]\rangle\;\;=\;\;log[\langle\rm
M_{H{\textsc{i}}}\rangle]-\frac{{\rm ln}\>10}{2}\sigma^{2}\;,$ (2)

where $\sigma$ is the logarithmic scatter of the Hi masses in the bin, with
$\sigma=0.26$ dex at $z\approx 0$ (Dénes et al., 2014). However, there exists
no estimate of the scatter of the $\rm M_{H{\textsc{i}}}-\rm M_{B}$ scaling
relation at $z\sim 0.35$. We hence assume that galaxies at $z\approx 0.35$
show the same scatter as galaxies in the nearby Universe, and use the value
$\sigma=0.26$ to estimate $\langle\rm log[\rm M_{H{\textsc{i}}}]\rangle$ for
different $\rm M_{B}$ bins. Later in this section, we show that $\sigma=0.26$
is consistent with the constraints on the scatter obtained from our full
sample of galaxies.

To measure the $\rm M_{H{\textsc{i}}}-\rm M_{B}$ scaling relation for galaxies
at $z\sim 0.35$, the values of $\langle{\rm log}\>\rm
M_{H{\textsc{i}}}\rangle$ in the three $\rm M_{B}$ bins of Table 1 were
estimated from Equation 2 by subtracting 0.078 dex from the measurements of
${\rm log}[\langle\rm M_{H{\textsc{i}}}\rangle]$. We note that this assumes a
log-normal distribution of the individual Hi masses with a scatter of $\approx
0.26$ dex in each $\rm M_{B}$ bin. A scaling relation of the parametric
form333We have chosen to normalize the scaling relation at the centre of the
$\rm M_{B}$ range of our sample, $\rm M_{B}=-19$, so as to obtain the lowest
covariance between the errors on the slope and the normalization of the
relation.

${\rm log}\left(\frac{\rm M_{H{\textsc{i}}}}{{\rm
M}_{\odot}}\right)=\alpha_{19}+\beta(\rm M_{B}+19)$ (3)

was then fitted to the estimates of $\langle{\rm log}[\rm
M_{H{\textsc{i}}}]\rangle$ at the median $\rm M_{B}$ values of Table 1, via a
chi-square minimization approach. The uncertainties associated with the best-
fit parameters were estimated from the error covariance matrix of the fit,
obtained using the jackknife standard errors as $1\sigma$ errors on the data
points. The best-fit $\rm M_{H{\textsc{i}}}-\rm M_{B}$ scaling relation for
blue galaxies at $z\approx 0.35$ is found to be

${\rm log}\left(\frac{\rm M_{H{\textsc{i}}}}{\rm M_{\odot}}\right)=(9.303\pm
0.068)-(0.184\pm 0.053)(\rm M_{B}+19)\,.$ (4)

Figure 2 shows the $\rm M_{H{\textsc{i}}}-\rm M_{B}$ scaling relation of
Equation 4, with, for comparison, the local $\rm M_{H{\textsc{i}}}-\rm M_{B}$
relation of Dénes et al. (2014). It is clear that the $\rm
M_{H{\textsc{i}}}-\rm M_{B}$ relation at $z\approx 0.35$ is significantly
flatter than the local relation, with the slopes being discrepant at $\approx
3\sigma$ significance. Our results indicate that luminous galaxies at
$z\approx 0.35$ are Hi-poor on average, compared to their local counterparts,
while fainter galaxies at $z\approx 0.35$ tend to contain more Hi than similar
galaxies at the present epoch.

For a symmetric logarithmic scatter around the $\rm M_{H{\textsc{i}}}-\rm
M_{B}$ scaling relation, $\langle{\rm log}[\rm M_{H{\textsc{i}}}]\rangle={\rm
log}[\rm M_{H{\textsc{i}}}^{\rm med}]$, where $\rm M_{H{\textsc{i}}}^{med}$ is
the median value of $\rm M_{H{\textsc{i}}}$. This implies that median-stacking
of the Hi 21 cm emission signals from the galaxies in each $\rm M_{B}$ bin can
be used to directly estimate $\langle{\rm log}[\rm M_{H{\textsc{i}}}]\rangle$.
However, in the current analysis, the detection significances of the median Hi
masses in the different $\rm M_{B}$ bins are too low to use this approach. We
hence used mean-stacking (which yields a higher, $>4\sigma$, detection
significance in each $\rm M_{B}$ bin) and Equation 2 to estimate $\langle{\rm
log}[\rm M_{H{\textsc{i}}}]\rangle$. If the median Hi masses in multiple $\rm
M_{B}$ bins could be measured with high significance, median-stacking would be
a better approach to robustly determine the Hi scaling relations.

While the sensitivity of the current Hi stacks is not sufficient to detect the
median Hi mass at sufficient significance in each $\rm M_{B}$ bin, we do
detect the median Hi 21 cm emission signal of the full sample of galaxies with
a high significance (see Section 3). We obtain a median Hi mass of $(1.17\pm
0.19)\times 10^{9}\ \rm M_{\odot}$, while the mean Hi mass of the full sample
is $(1.57\pm 0.22)\times 10^{9}\ \rm M_{\odot}$. We use these estimates of the
mean and the median Hi masses of the full sample to constrain the scatter
around the $\rm M_{H{\textsc{i}}}-\rm M_{B}$ scaling relation. For a log-
normal distribution for the Hi masses of the full sample (i.e. effectively
neglecting the slope of the scaling relation), the log-normal scatter can be
estimated from Equation 2, as

$\sigma_{\rm FS}={\rm\sqrt{(2/ln\ 10)\ log[\langle\rm
M_{H{\textsc{i}}}\rangle/\rm M_{H{\textsc{i}}}^{med}]}}\;,$ (5)

where we have used the fact that, for a log-normal distribution, $\rm\langle
log[\rm M_{H{\textsc{i}}}]\rangle=log[\rm M_{H{\textsc{i}}}^{med}]$. For our
full sample, we find $\rm\sigma_{FS}=0.33\pm 0.10$ dex. However, this is an
upper limit to the true scatter in the Hi mass around the scaling relation,
because a non-zero slope in the relation causes an additional spread in the Hi
mass, over and above the log-normal scatter. We hence used a Monte Carlo
approach to estimate the true scatter, combining the measured value of
$\rm\sigma_{FS}=0.33\pm 0.10$ dex with the known $\rm M_{B}$ distribution of
our sample. We randomly assigned Hi masses to galaxies in our sample based on
their $\rm M_{B}$ values and the measured slope of the scaling relation444Note
that the normalization of the scaling relation has no effect on this
analysis., assuming different values of the true scatter, and measured
$\rm\sigma_{FS}$ for each distribution using Equation 5. For the allowed range
$\rm\sigma_{FS}=0.33\pm 0.10$ dex, we obtain a true scatter of
$\sigma=0.21^{+0.14}_{-0.21}$ dex. We note that this estimate of the true
scatter is entirely consistent with the local value, $\sigma=0.26$ dex.

Finally, the DEEP2 and DEEP3 galaxy samples are complete to $\rm R_{AB}<24.1$,
corresponding to $\rm M_{B}<-18$ for our redshift range. This implies that the
faintest $\rm M_{B}$ bin, with $-18<\rm M_{B}\leq-16$, suffers from
incompleteness (Bera et al., in prep.), with the $\rm M_{B}$ distribution of
galaxies in this bin different from that of the true cosmological distribution
at $z\approx 0.35$. To test whether this incompleteness might introduce a bias
in the inferred parameters of the $\rm M_{H{\textsc{i}}}-\rm M_{B}$ scaling
relation, we carried out a Monte Carlo simulation to test whether a given
input scaling relation would be recovered via our approach, given the actual
$\rm M_{B}$ distribution of our galaxy sample. This was done by assuming a
range of $\rm M_{H{\textsc{i}}}-\rm M_{B}$ scaling relations of the form of
Equation 1, each with a logarithmic scatter of 0.26 dex. For each such scaling
relation, we assigned each galaxy in our sample a value of $\rm
M_{H{\textsc{i}}}$ based on its $\rm M_{B}$ value. We then measured the
average Hi mass in each of the three $\rm M_{B}$ bins of Table 1, determined
the value of $\langle{\rm log}[\rm M_{H{\textsc{i}}}]\rangle$ for each bin,
and carried out a least-squares fit to determine the best-fit $\rm
M_{H{\textsc{i}}}-\rm M_{B}$ scaling relation. The incompleteness of the $\rm
M_{B}$ sample was found to have no effect on the normalization of the $\rm
M_{H{\textsc{i}}}-\rm M_{B}$ relation; we hence fixed the value of
$\alpha_{19}$ to $9.30$. The slope of the input scaling relation, $\beta$, was
allowed to vary between $-0.34$ and 0, i.e. between the value in the local
Universe and a flat $\rm M_{H{\textsc{i}}}-\rm M_{B}$ relation. For all values
of $\beta$ in this range, the obtained best-fit scaling relation was found to
be consistent with the input scaling relation, within the measurement
uncertainties. We hence conclude that the $\rm M_{H{\textsc{i}}}-\rm M_{B}$
scaling relation estimated in this study, for which the $\beta$ value lies
within the simulated range, is not significantly affected by the
incompleteness of the galaxy sample.

##  5 The Hi mass function at $z\sim 0.35$

###  5.1 Estimating the HiMF: Validation in the local Universe

Figure 3: The HiMF at $z\approx 0$ inferred by combining the local $\rm
M_{H{\textsc{i}}}-\rm M_{B}$ relation with the B-band luminosity function,
with [A] no assumed scatter in the $\rm M_{H{\textsc{i}}}-\rm M_{B}$ relation,
and [B] a scatter of $\sigma=0.26$ dex, equal to that in the local Universe
(Dénes et al., 2014). In each panel, the blue stars show the HiMF of local
galaxies inferred here, while the solid line shows the best-fit Schechter
function to the inferred HiMF. The dashed line indicates the HiMF measured
from the unbiased ALFALFA survey (Jones et al., 2018). It is clear that the
assumption of zero scatter results in a significant under-estimate of the
number of high-mass galaxies, while the assumption $\sigma=0.26$ dex yields a
HiMF that is in good agreement with the directly measured one. Figure 4: [A]
The HiMF at $z\approx 0.35$, inferred by combining our $\rm
M_{H{\textsc{i}}}-\rm M_{B}$ relation with the ALHAMBRA B-band luminosity
function (López-Sanjuan et al., 2017). The blue stars represent the inferred
HiMF of blue galaxies at $z\sim 0.35$, while the dashed line shows the ALFALFA
HiMF at $z\approx 0$ (Jones et al., 2018). [B] The inferred number density of
blue star-forming galaxies at $z\approx 0.35$ above a given Hi mass, as a
function of $\rm log(\rm M_{H{\textsc{i}}})$. The dashed curve shows the same
quantity in the local Universe, from the HiMF of ALFALFA survey (Jones et al.,
2018).

The HiMF, $\Phi(\rm M_{H{\textsc{i}}})$, is defined as the number of galaxies
within a unit comoving cosmic volume having Hi masses between $\rm
M_{H{\textsc{i}}}$ and $\rm M_{H{\textsc{i}}}+{\rm d}\rm M_{H{\textsc{i}}}$
for a given Hi mass, $\rm M_{H{\textsc{i}}}$. This can be estimated either via
unbiased Hi 21 cm emission surveys (e.g. Zwaan et al., 2005; Jones et al.,
2018), or by combining the $\rm M_{H{\textsc{i}}}-\rm M_{B}$ scaling relation
with the B-band galaxy luminosity function (e.g. Rao & Briggs, 1993; Zwaan et
al., 2001). The early studies that followed the latter approach were
analytical in nature, neglecting the scatter in the $\rm M_{H{\textsc{i}}}-\rm
M_{B}$ relation. However, the scatter in this relation is critical to
correctly estimate the HiMF: Ignoring the scatter leads to under-estimating
the HiMF at the high-mass end, as we demonstrate below (see Fig. 3).

We followed a Monte Carlo approach to estimate the HiMF, combining the B-band
luminosity function and the $\rm M_{H{\textsc{i}}}-\rm M_{B}$ scaling
relation. We initially applied this approach to galaxies in the local
Universe, where the HiMF has been directly measured from unbiased Hi 21 cm
surveys. The B-band luminosity function at $z\approx 0$ is well described by
the Schechter parametrization (e.g. Schechter, 1976; Faber et al., 2007)

$\phi(L_{\rm B})=\left(\frac{\phi_{*}}{L_{\rm B}^{*}}\right)\left(\frac{L_{\rm
B}}{L_{\rm B}^{*}}\right)^{\alpha}e^{-L_{\rm B}/L_{\rm B}^{*}}\;,$ (6)

with $\alpha=-1.03$, $\phi_{*}=5.9\times 10^{-3}\>{\rm Mpc^{-3}}$ and $M_{\rm
B}^{*}=-20.04$ (Bell et al., 2003; Faber et al., 2007).

Using the above B-band luminosity function, we generated a sample of galaxies
with known absolute B-band magnitudes within a cosmic volume of $\rm
10^{7}\>{\rm Mpc}^{3}$. We assigned each galaxy a value of $\rm log(\rm
M_{H{\textsc{i}}})$ via the local $\rm M_{H{\textsc{i}}}-\rm M_{B}$ relation
of Equation 1 (Dénes et al., 2014). This was done via two approaches, [A]
assuming no scatter in the $\rm M_{H{\textsc{i}}}-\rm M_{B}$ relation, and [B]
assuming a logarithmic scatter of 0.26 dex, as measured in the local Universe
(Dénes et al., 2014). Finally, for each approach, we inferred a non-parametric
HiMF by counting the number of galaxies in logarithmic bins of Hi mass (with a
bin width of 0.2 dex), and dividing this number by the width of the bin and by
the total cosmic volume. The errors on the inferred HiMF were estimated from
$10^{4}$ realizations of the galaxy sample.

Figures 3[A] and [B] show the inferred local HiMF from our approach, for the
cases with zero assumed scatter and $\sigma=0.26$ dex, respectively, along
with the direct estimate of the local HiMF from the unbiased ALFALFA survey
(Jones et al., 2018). We note the $\rm M_{H{\textsc{i}}}-\rm M_{B}$ scaling
relation was measured for galaxies with $\rm M_{H{\textsc{i}}}>10^{9}\ \rm
M_{\odot}$, and hence restrict the figures to galaxies above this Hi mass. It
is clear from Fig. 3[A] that neglecting the scatter in the $\rm
M_{H{\textsc{i}}}-\rm M_{B}$ relation results in significantly under-
estimating the HiMF at the high-mass end. Conversely, Fig. 3[B] shows that
using a scatter of $0.26$ dex yields an HiMF that is in excellent agreement
with the ALFALFA HiMF.

Following the literature, we assume the following Schechter function form for
the HiMF

$\Phi(\rm M_{H{\textsc{i}}})=\frac{\Phi_{*}}{M_{0}^{*}}\left(\frac{\rm
M_{H{\textsc{i}}}}{M_{0}^{*}}\right)^{a}exp\left[-\left(\frac{\rm
M_{H{\textsc{i}}}}{M_{0}^{*}}\right)\right]\;.$ (7)

The best-fit Schechter parameters for our inferred HiMF are $\Phi_{*}=(4.0\pm
0.4)\times 10^{-3}\>{\rm Mpc^{-3}}$, $a=-1.29\pm 0.03$, and ${\rm
log}(M_{0}^{*}/M_{\odot})=9.94\pm 0.03$. These are all in good agreement with
the ALFALFA estimates of the same parameters, $\Phi_{*}=(4.5\pm 0.2)\times
10^{-3}\>{\rm Mpc^{-3}}$, $a=-1.25\pm 0.02$, and ${\rm
log}(M_{0}^{*}/M_{\odot})=9.94\pm 0.01$ (Jones et al., 2018).

We conclude that it is indeed possible to determine the HiMF by combining the
B-band luminosity function with the $\rm M_{H{\textsc{i}}}-\rm M_{B}$
relation, but that a critical ingredient in the estimate is the scatter in the
latter relation. Not including this scatter results in an under-estimate of
the HiMF at the high-mass end, while over-estimating the scatter yields an
over-estimate of the number of high-mass galaxies. Thus, an accurate estimate
of the scatter in the $\rm M_{H{\textsc{i}}}-\rm M_{B}$ relation is crucial
for an estimate of the HiMF via this approach.

###  5.2 Determining the HiMF at $z\approx 0.35$

Having confirmed that combining the B-band luminosity function and the $\rm
M_{H{\textsc{i}}}-\rm M_{B}$ scaling relation yields the expected HiMF at
$z\approx 0$, we used the same approach to estimate the HiMF at $z\approx
0.35$ for the blue galaxies of our EGS survey. For this, we used the B-band
luminosity function of star-forming galaxies at $0.2<z<0.4$ from the ALHAMBRA
survey555Similar results were obtained on using the B-band luminosity function
of blue galaxies at $z\approx 0.35$ from the DEEP2 survey (Willmer et al.,
2006). ($\alpha=-1.29$, $\phi_{*}=3.1\times 10^{-3}\>{\rm Mpc^{-3}}$ and
$M_{\rm B}^{*}=-20.85$; López-Sanjuan et al., 2017). We note that the B-band
luminosity function at $z\approx 0.3$ is different from that in the local
Universe, with a higher number density of luminous galaxies, with $M_{\rm
B}<-20$, at $z\approx 0.3$. The measured $\rm M_{H{\textsc{i}}}-\rm M_{B}$
scaling relation from the present work (Equation 4) was used to assign Hi
masses to the galaxies within a volume of $10^{7}\>{\rm Mpc}^{3}$, again
assuming a logarithmic scatter of 0.26 dex. The errors on the inferred HiMF
were estimated from $10^{4}$ realizations of the galaxy sample via a Monte
Carlo approach, taking into account the errors on the parameters of the $\rm
M_{H{\textsc{i}}}-\rm M_{B}$ scaling relation of Equation 4.

Figure 4[A] shows the inferred HiMF for blue galaxies at $z\approx 0.35$; for
comparison, the dashed line shows the ALFALFA HiMF at $z\approx 0$. Note that
we only considered galaxies with $\rm M_{B}\leq-16$ in our analysis, the $\rm
M_{B}$ range over which we measured the $\rm M_{H{\textsc{i}}}-\rm M_{B}$
relation. The relatively large uncertainties on the inferred HiMF do not allow
us to obtain robust estimates of the best-fit Schechter parameters. However,
Figure 4[A] clearly indicates that there are far fewer massive galaxies at
$z\approx 0.35$, with $\rm M_{H{\textsc{i}}}\gtrsim 10^{10}\;\rm M_{\odot}$,
than in the local Universe. Figure 4[B] shows the number density of galaxies
above a given Hi mass at $z\approx 0.35$ (blue stars) and at $z\approx 0$
(dashed curve) as a function of $\rm log[\rm M_{H{\textsc{i}}}]$. The number
density of blue galaxies666Note that the uncertainties associated with the
number density at $z\approx 0$ are dominated by the conservative estimate of
cosmic variance of Jones et al. (2018), obtained by comparing results from the
ALFALFA Spring and Fall samples. with $\rm M_{H{\textsc{i}}}>10^{10}\;\rm
M_{\odot}$ is $(6.8\pm 1.2)\times 10^{-4}\;{\rm Mpc}^{-3}$ at $z\approx 0$
(assuming the ALFALFA HiMF; Jones et al., 2018) but only $(2.0\pm 1.7)\times
10^{-4}\;{\rm Mpc}^{-3}$ at $z\approx 0.35$. We thus find evidence that the
cosmological number density of galaxies with high Hi masses, $\rm
M_{H{\textsc{i}}}>10^{10}\;\rm M_{\odot}$ at $z\approx 0.35$ is smaller by a
factor of $\approx 3.4$ than in the local Universe. Finally, we note that
Figure 4[A] also indicates that there are more “low-mass” galaxies with Hi
masses $\approx 10^{9}\;\rm M_{\odot}$ at $z\approx 0.35$ than in the local
Universe. Interestingly enough, these results are both broadly consistent with
predictions from the GALFORM semi-analytical model (Baugh et al., 2019).

The fact that the cosmic number density of galaxies with a high Hi mass, $\rm
M_{H{\textsc{i}}}\gtrsim 10^{10}\;\rm M_{\odot}$, increases significantly from
$z\approx 0.35$ to the present epoch is directly linked to the relatively low
Hi content of luminous galaxies at $z\approx 0.35$ compared to the local
Universe (see Section 4). Our results indicate that massive, luminous galaxies
at $z\approx 0$ have acquired a significant amount of Hi over the past four
Gyr, through either merger events or accretion from the CGM.

It should be emphasized that the cosmic volume of the EGS covered in our uGMRT
observations, $\approx 4.7\times 10^{4}$ comoving Mpc3, is relatively small.
Our results could hence be affected by cosmic variance (e.g. Driver &
Robotham, 2010). For example, some of our galaxies may reside in dense
environments like groups or clusters (e.g. Gerke et al., 2012), which could
affect their Hi content (e.g. via ram-pressure stripping; Gunn & Gott, 1972;
Chung et al., 2009). We have found no direct evidence for the presence of
galaxy clusters within our target volume, but cannot rule out the possibility
of galaxy groups. Similar studies covering a larger cosmic volume would be
useful to test the results of the present survey.

##  6 Summary

We report the first measurement of the $\rm M_{H{\textsc{i}}}-\rm M_{B}$
scaling relation at cosmological distances, $z\approx 0.35$, using the
technique of spectral line stacking applied to a deep GMRT Hi 21 cm emission
survey of the EGS at $z\approx 0.2-0.42$. We obtain the relation ${\rm
log}\left(\rm M_{H{\textsc{i}}}/\rm M_{\odot}\right)=(9.303\pm
0.068)-(0.184\pm 0.053)(\rm M_{B}+19)$, for blue star-forming galaxies at
$z\approx 0.35$, significantly flatter than the corresponding relation in the
local Universe. This implies that faint star-forming galaxies at $z\approx
0.35$ have a higher Hi mass than similar galaxies at $z\approx 0$, while
luminous galaxies have a lower Hi mass than their local counterparts. We
combined this $\rm M_{H{\textsc{i}}}-\rm M_{B}$ scaling relation (with a
scatter assumed to be equal to that in the local Universe) with the known
B-band luminosity function of star-forming galaxies at $z=0.2-0.4$ to
determine the Hi mass function at $z\approx 0.35$. We find that the
cosmological number density of galaxies with $\rm M_{H{\textsc{i}}}\gtrsim
10^{10}\;\rm M_{\odot}$ at $z\approx 0.35$ is smaller by a factor of $\approx
3.4$ than at the present epoch. Conversely, there appear to be more galaxies
with low Hi masses, $\rm M_{H{\textsc{i}}}\approx 10^{9}\ \rm M_{\odot}$, at
$z\approx 0.35$ than at $z\approx 0$. While the comoving volume covered by our
survey is relatively small, implying that cosmic variance may be an issue, our
results indicate that luminous, massive galaxies in the local Universe have
acquired a significant amount of Hi over the last $\approx 4$ Gyr, via either
mergers or accretion.

We thank the staff of the GMRT who have made these observations possible. The
GMRT is run by the National Centre for Radio Astrophysics of the Tata
Institute of Fundamental Research. AB and NK thank Aditya Chowdhury and
Balpreet Kaur for detailed comments on an earlier draft of the manuscript and
many discussions on Hi 21 cm stacking that have contributed to this paper. NK
acknowledges support from the Department of Science and Technology via a
Swarnajayanti Fellowship (DST/SJF/PSA-01/2012-13). AB, NK, $\&$ JNC also
acknowledge the Department of Atomic Energy for funding support, under project
12-R&D-TFR-5.02-0700.

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