Central set Theorem near zero

E. Bayatmanesh

M. Akbari Tootkaboni

Department of Mathematics, Faculty of Basic Science,

Shahed University,

Tehran-Iran.

E-mail<EMAIL_ADDRESS>
E-mail<EMAIL_ADDRESS>
A. Bagheri Sales

Department of Mathematics, Faculty of science,

Qom University,

Qom-Iran.

E-mail: alireza bagheri<EMAIL_ADDRESS>

> Abstract. In this paper, we introduce notions of $J$-set near zero and
> $C$-set near zero for a dense subsemigroup of $((0,+\infty),+)$ and obtain
> some results for them. Also we derive the Central Sets Theorem near zero.

Keywords: Central set Theorem, The Stone-Cech compactification, $C$-set,
$J$-set, Piecewise syndetic set near zero.

2000 Mathematics subject classification:  Primary: 54D80, 22A15, Secondary:
22A20.

##  1\. Introduction

Let $(S,+)$ be a discrete semigroup. The collection of all ultrafilters on $S$
is called the Stone-$\check{C}$ech compactification of $S$ and denoted by
$\beta S$. For $A\subseteq S$, define $\overline{A}=\\{p\in\beta S:A\in p\\}$,
then $\\{\overline{A}:A\subseteq S\\}$ is a basis for the open sets( also for
the closed sets) of $\beta S$. There is a unique extension of the operation to
$\beta S$ making $(\beta S,+)$ a right topological semigroup ( i.e. for each
$p\in\beta S$, the right translation $\rho_{p}$ is continuous where
$\rho_{p}(q)=q+p$) and also for each $x\in S$, the left translation
$\lambda_{x}$ is continuous where $\lambda_{x}(q)=x+q$. The principal
ultrafilters being identified with the points of $S$ and $S$ is a dense subset
of $\beta S$. Given $p,q\in\beta S$ and $A\subseteq S$, we have $A\in p+q$ if
and only if $\\{x\in S:-x+A\in q\\}\in p$, where $-x+A=\\{y\in S:x+y\in A\\}$.

A nonempty subset $L$ of a semigroup $(S,+)$ is called a left ideal of $S$ if
$S+L\subseteq L$, a right ideal if $L+S\subseteq L$, and a two sided ideal (or
simply an ideal ) if it is both a left and right ideal. A minimal left ideal
is a left ideal that does not contain any proper left ideal. Similarly, we can
define minimal right ideal and smallest ideal.

Any compact Hausdorff right topological semigroup $(S,+)$ has a smallest two
sided ideal, denoted by $K(S)$,which is the union of all minimal left ideals.
$K(S)$ is the union of all minimal right ideals, as well. Given a minimal left
ideal $L$ and a minimal right ideal $R$, $L\cap R$ is a group, and in
particular contains an idempotent. An idempotent in $K(S)$ is called a minimal
idempotent. An idempotent is minimal if and only if it is a member of the
smallest ideal. For more details see [7].

For $A\subseteq S$, and $p\in\beta S$, we define $A^{*}(p)=\\{s\in A:-s+A\in
p\\}$.

######  Lemma 1.1.

Let $(S,+)$ be a semigroup, let $p+p=p\in\beta S$, and let $A\in p$. For each
$s\in A^{*}(p)$, $-s+A^{*}(p)\in p$.

###### Proof.

[7], Lemma 4.14. ∎

Now we review the definition of partition regularity and a theorem that
connects it with ultrafilters.

######  Definition 1.1.

Let $\mathcal{R}$ be a nonempty set of subsets of $S$. $\mathcal{R}$ is
partition regular if and only if whenever $\mathcal{F}$ is a finite set of
$\mathcal{P}(S)$ and $\bigcup\mathcal{F}\in\mathcal{R}$, there exist
$A\in\mathcal{F}$ and $B\in\mathcal{R}$ such that $B\subseteq A$.

######  Theorem 1.2.

Let $\mathcal{R}\subseteq\mathcal{P}(S)$ be a nonempty set and assume
$\emptyset\notin\mathcal{R}$. Let

$\mathcal{R}^{\uparrow}=\\{B\in\mathcal{P}(S):A\subseteq B\mbox{ for some
}A\in\mathcal{R}\\}.$

Then (a), (b) and (c) are equivalence.

(a) $\mathcal{R}$ is partition regular.

(b) Whenever $\mathcal{A}\subseteq\mathcal{P}(S)$ has the property that every
finite nonempty subfamily of $\mathcal{A}$ has an intersection which is in
$\mathcal{R}^{\uparrow}$, there is $\mathcal{U}\in\beta S_{d}$ such that
$\mathcal{A}\subseteq\mathcal{U}\subseteq\mathcal{R}^{\uparrow}$.

(c) Whenever $A\in\mathcal{R}$, there is $\mathcal{U}\in\beta S_{d}$ such that
$A\in\mathcal{U}\subseteq\mathcal{R}^{\uparrow}$.

Proof.

[7, Theorem 3.11].$\hfill\square$

######  Definition 1.3.

Let $(S,.)$ be a discrete semigroup, let $A\subseteq S$. Then $A$ is a central
set if and only if there exists an idempotent $p$ in the smallest ideal of
$\beta S$ with $A\in p$.

In this paper, the collection of all nonempty subsets of $S$ is denoted by
$P_{f}(S)$ and $P(S)$ is the set of all subsets of $S$. Central subsets of a
discrete semigroup $S$ have very strong combinatorial properties which are
consequences of the central set theorem. There is an elementary description of
central sets which was showed in [7].

######  Theorem 1.4 (Central Set Theorem [1]).

Let $(S,+)$ be a commutative subsemigroup and let
$\mathcal{T}={}^{\mathbb{N}}S$, the set of sequences in $S$. Let $A$ be a
central subset of $S$. There exist functions
$\alpha:P_{f}(\mathcal{T})\rightarrow S$ and $H:P_{f}(\mathcal{T})\rightarrow
P_{f}(\mathbb{N})$ such that

(1) if $F,G\in P_{f}(\mathcal{T})$ and $F\subsetneq G$, then
$maxH(F)<minH(G)$, and

(2) whenever $m\in\mathbb{N}$, $G_{1},\cdots,G_{m}\in P_{f}(\mathcal{T})$,
$G_{1}\subsetneq G_{2}\subsetneq\cdots\subsetneq G_{m}$, and for each
$i\in\\{1,2,\cdots,m\\}$, $f_{i}\in G_{i}$, one has

$\sum_{i=1}^{m}\big{(}\alpha(G_{i})+\sum_{t\in H(G_{i})}f_{i}(t)\big{)}\in A.$

We define a set to be a $C-$set if and only if it satisfies the conclusion of
Central Set Theorem.

######  Definition 1.5.

Let $S$ be a commutative subsemigroup and let $A\subseteq S$ and let
$\mathcal{T}=^{\mathbb{N}}S$. The set $A$ is a $C-$set if and only if there
exist functions $\alpha:P_{f}(\mathcal{T})\rightarrow S$ and
$H:P_{f}(\mathcal{T})\rightarrow P_{f}(\mathbb{N})$ such that

(1) if $F,G\in P_{f}(\mathcal{T})$ and $F\subsetneq G$, then
$maxH(F)<minH(G)$, and

(2) whenever $m\in\mathbb{N}$, $G_{1},\cdots,G_{m}\in P_{f}(\mathcal{T})$,
$G_{1}\subsetneq G_{2}\subsetneq\cdots\subsetneq G_{m}$, and for each
$i\in\\{1,2,\cdots,m\\}$, $f_{i}\in G_{i}$, one has

$\sum_{i=1}^{m}\big{(}\alpha(G_{i})+\sum_{t\in H(G_{i})}f_{i}(t)\big{)}\in A.$

The central sets are important, and algebraically are easy to work with.
However, from a combinatorial viewpoint, $C-$sets are the objects that matter.
In $(\mathbb{N},+)$ and in many other semigroups, they are the objects that
contain solutions to partition regular systems of homogeneous equations as
well as the other myriads of properties that are consequences of Central Sets
Theorem, see e.g. [9]. In [1] authors obtained a simple characterization of
$C-$sets in an arbitrary discrete semigroup.

######  Definition 1.6.

Let $(S,+)$ be a commutative subsemigroup and let
$\mathcal{T}=^{\mathbb{N}}S$. Let $A\subseteq S$ is a $J-$set if and only if
whenever $F\in P_{f}(\mathcal{T})$, there exist $a\in S$ and $H\in
P_{f}(\mathbb{N})$ such that for each $f\in F$, $a+\sum_{t\in H}f(t)\in A$.

If $S$ is noncommutative, then the definition of $J-$sets is somewhat more
complicated, but still much simpler than the statement of the noncommutative
Central Sets Theorem. One of the reasons that $J-$sets are of interest is
that, if $S$ is a discrete commutative semigroup, then every subset of $S$
with positive upper density is a $J-$set, see [9].

In [6], N. Hindman and I. Leader investigated some concepts near zero. The set
$0^{+}$ of all non principal ultrafilters on $S=((0,\infty),+)$ that is
convergent to $0$ is a semigroup under the restriction of the usual
${}^{\prime}+^{\prime}$ on $\beta S$, the Stone-$\check{C}$Cech
compactification of the discrete semigroup $S=((0,\infty),+)$. Further, in [2]
and [8], De, Hindman and Strauss worked out some more and much more elaborated
applications of central sets. One such property is that in any finite
partition of a central set at least one cell of the partition is central set
[5]. In [2] the authors used the algebraic structure of $0^{+}(S)$ in their
investigation of image partition regularity near $0$ of finite and infinite
matrices.

In [4] the algebraic structure of $0^{+}(\mathbb{R})$ is used to investigate
image partition regularity of matrices with real entries form $\mathbb{R}$.
Central sets near zero were introduced by N. Hindman and I. Learder in [6] as
central sets, central sets near zero enjoy rich combinatorial structure ,too.
Central sets are ideal objects for Ramsey theoretic. The central sets theorem
was first introduced by H. Furstenberg ( see [5]) for the semigroup
$\mathbb{N}$ and considering sequence in $\mathbb{Z}$. The most general
version of Central Sets Theorem is available in [1].

We have been considering semigroups which are dense in $((0,\infty),+)$ with
natural topology. When passing to the Stone-$\check{C}$ech compactification of
such a semigroup S, we deal with $S_{d}$, which is the set $S$ with the
discrete topology.

######  Definition 1.7.

Let $S$ be a dense subset of $((0,\infty),+)$. Then

$0^{+}(S)=\\{p\in\beta S_{d}:(\forall\epsilon>0)\,\,(0,\epsilon)\cap S\in
p\\}.$

By Lemma 2.5 in [6], $0^{+}(S)$ is a compact right topological subsemigroup of
$(\beta S_{d},+)$, and $0^{+}(S)\cap K(\beta S_{d})=\emptyset$. Since
$0^{+}(S)$ is compact right topological semigroup, so $0+(S)$ contains minimal
idempotents.

In section 2 we difine notions of thick near zero, syndetic near zero and
piecewise syndetic near zero and we obtain a main theorem about piecewise
syndetic near zero.

In section 3, we introduce Central Set Theorem near zero, also we define
$C$-set near zero, $J$-set near zero and derive some results for dense
subsemigroups of $((0,\infty),+)$. We will be considering semigroups which are
dense with respect to the usual topology on $((0,\infty),+)$.

##  2\. Additive properties near zero

In this section, we define the notions of thick near zero, syndetic near zero
and piecewise syndetic near zero, which in turn extend the notions of thick,
syndetic and piecewise syndetic. Recall that $A\subseteq S$ is thick if and
only if for every $F\in\mathcal{P}_{f}(S)$ there exists $x\in S$ such that
$F+x\subseteq A$, $A$ is syndetic if and only if there exists
$H\in\mathcal{P}_{f}(S)$ such that $S=\bigcup_{t\in H}-t+A$ and $A$ is
piecewise syndetic if and only if there exists $H\in\mathcal{P}_{f}(S)$ such
that $\bigcup_{t\in H}-t+A$ is thick.

######  Definition 2.1.

Let $S$ be a dense subsemigroup of $((0,\infty),+)$, and let $A\subseteq S$.

$(a)$ $A$ is thick near zero if and only if $(\exists\epsilon>0)(\forall
F\in(P_{f}(0,\epsilon)\cap S))(\forall\delta>0)(\exists y\in(0,\delta)\cap
S)(F+y\subseteq A)$.

$(b)$ $A$ is syndetic near zero if and only if for any $\epsilon>0$ there
exist $F\in\mathcal{P}_{f}((0,\epsilon)\cap S)$ and $\delta>0$ such that
$(0,\delta)\cap S\subseteq\bigcup_{t\in F}-t+A$.

$(c)$ $A$ is piecewise syndetic near zero if and only if for all $\delta>0$
there exists $F\in\mathcal{P}_{f}((0,\delta)\cap S)$ such that $\bigcup_{t\in
F}-t+A$ be thick near zero.

$(d)$ $A$ is a central set near zero if and only if there exists an idempotent
$p$ in the smallest ideal of $0^{+}(S)$ with $A\in p$.

######  Remark 2.2.

It is obvious that $A$ is thick near zero if and only if for some $\delta>0$

$\\{-t+A:t\in(0,\delta)\cap S\\}$

has the finite intersection property in $(0,\delta)\cap S$, i.e. for each
$t_{1},\cdots,t_{n}\in(0,\delta)\cap S$ there exists $x\in(0,\delta)\cap S$
such that $x\in\bigcap_{i=1}^{n}-t_{i}+A$.

In this paper, the minimal ideal in $0^{+}(S)$ is denoted by $K$.

######  Theorem 2.3.

Let $S$ be a dense subsemigroup of $((0,+\infty),+)$ and let $p\in 0^{+}(S)$.
The following statements are equivalent.

(a) $p\in K$.

(b) For all $A\in p$, $\left\\{x\in S:-x+A\in p\right\\}$ is syndetic near
$0$.

(c) For all $r\in 0^{+}(S)$, $p\in 0^{+}(S)+r+p$.

###### Proof.

See [6] or see Theorem 3.4 in [10]. ∎

######  Theorem 2.4.

Let $A\subseteq S$. Then $K\cap\overline{A}\neq\emptyset$ if and only if $A$
is piecewise syndetic near $0$.

###### Proof.

Necessity. Pick $p\in K\cap\overline{A}$ and let $B=\\{x\in S:-x+A\in p\\}$.
By Theorem 2.3, $B$ is syndetic near $\mu$. So for every $\varepsilon>0$ there
exist $F\in\mathcal{P}_{f}((0,\varepsilon)\cap S)$ and $\delta>0$ such that
$(0,\delta)\cap S\subseteq\bigcup_{t\in F}-t+B$. So for each
$x\in(0,\delta)\cap S$, there exists $t\in F$ such that $x\in-t+B$, and so
$-(x+t)+A\in p$. Thus $-x+(\bigcup_{t\in F}-t+A)\in p$ and since
$(0,\delta)\cap S\in p$ for each $\delta>0$, hence $\\{-x+(\bigcup_{t\in
F}-t+A):x\in(0,\delta)\cap S\\}$ has the finite intersection property in
$(0,\delta)\cap S$. So by Remark 2.2, $A$ is piecewise syndetic near zero.

Sufficiency. Let $A$ be piecewise syndetic near zero. So for each
$n\in\mathbb{N}$ there exists $F_{n}\in\mathcal{P}_{f}((0,\frac{1}{n})\cap S)$
such that for some $\epsilon_{n}>0$ and for each
$G_{n}\in\mathcal{P}_{f}((0,\epsilon_{n})\cap S)$, for every $\delta_{n}$
there exists $y_{n}\in\mathcal{P}_{f}((0,\delta_{n})\cap S)$ such that
$G_{n}+y_{n}\subseteq\bigcup_{t\in F_{n}}(-t+A)$. Pick
$\delta_{n}=min({\dfrac{1}{n},\epsilon_{n}})$ For $G\in\mathcal{P}_{f}(S)$ and
$\mu>0$, let

$C(G,\mu)=\\{x\in(0,\mu):\forall
n\in\mathbb{N},G\cap(0,\delta_{n})+x\subseteq\bigcup_{t\in F_{n}}(-t+A)\\}.$

It is obvious that each $C(G,\mu)\neq\emptyset$. Also,

$C(G_{1}\cup G_{2},min\\{\mu_{1},\mu_{2}\\})\subseteq C(G_{1},\mu_{1})\cap
C(G_{2},\mu_{2}),$

for each $G_{1},G_{2}\in\mathcal{P}_{f}(S)$ and for each $\mu_{1},\mu>0$.
Therefore

$\\{C(G,\mu):G\in\mathcal{P}_{f}(S)\mbox{ and }\mu>0\\}$

has the finite intersection property, so pick $p\in\beta S_{d}$ with

$\\{C(G,\mu):G\in\mathcal{P}_{f}(S)\mbox{ and }\mu>0\\}\subseteq p.$

Since $C(G,\mu)\subseteq(0,\mu)\cap S$, so $p\in 0^{+}(S)$.

Now we claim that for each $n\in\mathbb{N}$, $0^{+}(S)+p\subseteq cl_{\beta
S_{d}}\bigcup_{t\in F_{n}}(-t+A)$, so let $n\in\mathbb{N}$ and let $q\in
0^{+}(S)$. To show that $\bigcup_{t\in F_{n}}(-t+A)\in q+p$, we show that

$(0,\delta_{n})\cap S\subseteq\\{y>0:-y+\bigcup_{t\in F_{n}}(-t+A)\in p\\}.$

So let $y\in(0,\delta_{n})\cap S$. Then $C(\\{y\\},\delta_{n})\in p$ and
$C(\\{y\\},\delta_{n})\subseteq-y+\bigcup_{t\in F_{n}}(-t+A)$. Now pick
$r\in(0^{+}(S)+p)\cap K$( since $0^{+}(S)+p$ is a left ideal of $0^{+}(S)$).
Given $n\in\mathbb{N}$, $\bigcup_{t\in F_{n}}(-t+A)\in r$ so pick $t_{n}\in
F_{n}$ such that $-t_{n}+A\in r$. Since for each $n\in\mathbb{N}$, $t_{n}\in
F_{n}\subseteq(0,\frac{1}{n})\cap S$ so $lim_{n\rightarrow\infty}t_{n}=0$. Now
pick $q\in 0^{+}(S)\cap cl_{\beta S_{d}}\\{t_{n}:n\in\mathbb{N}\\}$. Then
$q+r\in K$ and $\\{t_{n}:n\in\mathbb{N}\\}\subseteq\\{t\in S:-t+A\in r\\}$ so
$A\in q+r$. ∎

######  Corollary 2.5.

Let $(S,+)$ be a dense subsemigroup of $(0,\infty)$, piecewise syndeticity
near zero is partition regular.

###### Proof.

It is obvuous. ∎

##  3\. Central Sets

Central subsets of a discrete semigroups have very strong combinatorial
properties which are a consequence of the Central Sets Theorem.

######  Definition 3.1.

Let $S$ be a dense subsemigroup of $\big{(}(0,\infty),+\big{)}$. We say that
$f:\mathbb{N}\rightarrow S$ is near zero if $inff(\mathbb{N})=0$. The
collection of all functions that is near zero is denoted by $\mathcal{T}_{0}$.

######  Definition 3.2.

Let $S$ be a dense subsemigoup of $((0,\infty),+)$ and let $A\subseteq S$.
Then $A$ is a $J-$set near zero if and only if whenever $F\in
P_{f}(\mathcal{T}_{0})$ and $\delta>0$, there exist $a\in S\cap(0,\delta)$ and
$H\in P_{f}(\mathbb{N})$ such that for each $f\in F$, $a+\sum_{t\in H}f(t)\in
A$.

Of course, we can say that $A\subseteq S$ is a $J-$set near zero if and only
if for each $F\in P_{f}(\mathcal{T}_{0})$ and for each $\delta>0$, there exist
$a\in S$ and $H\in P_{f}(\mathbb{N})$ such that $a+\sum_{t\in H}f(t)\in
A\cap(0,\delta)$ for each $f\in F$, i.e. for each $\delta>0$,
$A\cap(0,\delta)$ is a $J-$set. It is obvious that every $J-$set near zero
respect to this definition is a $J-$set near zero by Definition 5.2. So we
focuse on Definition 5.2.

######  Lemma 3.3.

Let $S$ be a dense subsemigroup of $\big{(}(0,\infty),+\big{)}$ and let
$A\subseteq S$ be a $J$-set near zero. Whenever $m\in\mathbb{N}$ and $F\in
P_{f}(\mathcal{T}_{0})$ and $\delta>0$, there exist $a\in S\cap(0,\delta)$ and
$H\in P_{f}(\mathbb{N})$ such that $minH>m$ and for each $f\in F$,
$a+\sum_{t\in H}f(t)\in A$.

###### Proof.

See Lemma 14.8.2 in [7]. ∎

######  Theorem 3.4.

Let $S$ be a dense subsemigroup of $\big{(}(0,\infty),+\big{)}$ and let $A$ be
a subset of $S$. If $A$ is a piecewise syndetic near zero, then $A$ is a
$J$-set near zero.

###### Proof.

Let $F\in P_{f}(\mathcal{T}_{0})$, let $l=|F|$, and enumerate $F$ as
$\\{f_{1},\cdots,f_{l}\\}$. Let $Y=\prod_{t=1}^{l}0^{+}$. Then by Theorem 2.22
in [7], $Y$ is a compact right topological semigroup and if
$s\in\prod_{t=1}^{l}S$, then $\lambda_{s}$ is continuous. For $i\in\mathbb{N}$
and $\delta>0$, let

$I_{i,\delta}=\big{\\{}\big{(}a+\sum_{t\in H}f_{1}(t),\cdots,a+\sum_{t\in
H}f_{l}(t)\big{)}:a\in S\cap(0,\delta),\\\ H\in P_{f}(\mathbb{N}),\mbox{ and
}minH>i\big{\\}}\\\ $

and let $E_{i,\delta}=I_{i,\delta}\cup\\{(a,\cdots,a):a\in
S\cap(0,\delta)\\}$.

Let $E=\bigcap_{i\in\mathbb{N},\delta>0}\overline{E_{i,\delta}}$ and let
$I=\bigcap_{i\in\mathbb{N},\delta>0}\overline{I_{i,\delta}}$. It is obvious
that $E\subseteq 0^{+}$ and $I\subseteq 0^{+}$. We claim that $E$ is a
subsemigroup of $Y$ and $I$ is an ideal of $E$. To this end, let $p,q\in E$.
We show that $p+q\in E$ and if either $p\in I$ or $q\in I$, then $p+q\in I$.
Pick $\delta>0$, then $U=cl_{\beta S_{d}}\big{(}(0,\delta)\cap S\big{)}$ is an
open neighborhood of $p+q$ and let $i\in\mathbb{N}$. Since $\rho_{q}$ is
continuous, pick a neighborhood $V$ of $p$ such that $V+q\subseteq U$. Pick
$x\in E_{i,\frac{\delta}{3}}\cap V$ with $x\in I_{i,\frac{\delta}{3}}$ if
$p\in I$. If $x\in I_{i,\frac{\delta}{3}}$ so that $x=(a+\sum_{t\in
H}f_{1}(t),\cdots,a+\sum_{t\in H}f_{l}(t))$ for some $a\in
S\cap(0,\frac{\delta}{3})$ and some $H\in P_{f}(\mathbb{N})$ with $minH>i$,
let $j=maxH$. Otherwise, let $j=i$. Since $\lambda_{x}$ is continuous, pick a
neighborhood $W$ of $q$ such that $x+W\subseteq U$. Pick $y\in
E_{j,\frac{\delta}{3}}\cap W$ with $y\in I_{j,\frac{\delta}{3}}$ if $q\in I$.
Then $x+y\in E_{i,\delta}\cap U$ and if either $p\in I$ or $q\in I$, then
$x+y\in I_{i,\delta}\cap U$.

By Theorem 2.23 in [7], $K(Y)=\prod_{t=1}^{l}K(0^{+})$. Pick by Theorem 2.4
some $p\in K(0^{+})\cap\overline{A}$. Then $\overline{p}=(p,\cdots,p)\in
K(Y)$. We claim that $\overline{p}\in E$. To see this, let $U$ be a
neighborhood of $\overline{p}$, let $i\in\mathbb{N}$, and pick
$C_{1},\cdots,C_{l}\in p$ such that $\prod_{t=1}^{l}\overline{C_{t}}\subseteq
U$. Pick $a\in\bigcap_{t=1}^{l}C_{t}$. Then $\overline{a}=(a,\cdots,a)\in
U\cap E_{i,\delta}$. Thus $\overline{p}\in K(Y)\cap E$ and consequently
$K(Y)\cap E\neq\emptyset$. Then by Theorem 1.65 in [7], we have that
$K(E)=K(Y)\cap E$ and so $\overline{p}\in K(E)\subseteq I$. Then
$I_{1,\delta}\cap\prod_{t=1}^{l}\overline{A}\neq\emptyset$ for each
$\delta>0$, so pick $z\in I_{1,\delta}\cap\prod_{t=1}^{l}\overline{A}$ and
pick $a\in S\cap(0,\delta)$ and $H\in P_{f}(\mathbb{N})$ such that

$z=\big{(}a+\sum_{t\in H}f_{1}(t),\cdots,a+\sum_{t\in H}f_{l}(t)\big{)}.$

∎

######  Theorem 3.5 (Central set Theorem near zero).

Let $S$ be a dense subsemigroup of $\big{(}(0,\infty),+\big{)}$. Let $A$ be a
central subset of $S$ near zero. Then for each $\delta\in(0,1)$, there exist
functions $\alpha_{\delta}:P_{f}(\mathcal{T}_{0})\rightarrow S$ and
$H_{\delta}:P_{f}(\mathcal{T}_{0})\rightarrow P_{f}(\mathbb{N})$ such that
(1) $\alpha_{\delta}(F)<\delta$ for each $F\in P_{f}(\mathcal{T}_{0})$,
(2) if $F,G\in P_{f}(\mathcal{T}_{0})$ and $F\subset G$, then
$maxH_{\delta}(F)<minH_{\delta}(G)$ and
(3) whenever $m\in\mathbb{N}$, $G_{1},\cdots,G_{m}\in P_{f}(\mathcal{T}_{0})$,
$G_{1}\subset G_{2}\subset\cdots\subset G_{m}$, and for each
$i\in\\{1,2,\cdots,m\\}$, $f_{i}\in G_{i}$, one has

$\sum_{i=1}^{m}\big{(}\alpha_{\delta}(G_{i})+\sum_{t\in
H_{\delta}(G_{i})}f_{i}(t)\big{)}\in A.$

###### Proof.

Pick a minimal idempotent $p$ of $0^{+}$ such that $A\in p$. Let
$A^{*}=\\{x\in A:-x+A\in p\\}$, so $A^{*}\in p$. Also by Lemma 4.14 in [7], if
$x\in A^{*}$, then $-x+A^{*}\in p$.

We define $\alpha_{\delta}(F)\in S$ and $H_{\delta}(F)\in P_{f}(\mathbb{N})$
for $F\in P_{f}(\mathcal{T}_{0})$ by induction on $|F|$ satisfying the
following inductive hypotheses:
(1) $\alpha_{\delta}(G)<\delta$ for each $G\in\mathcal{T}_{0}$,
(2) if $F,G\in P_{f}(\mathcal{T}_{0})$ and $F\subset G$, then
$maxH_{\delta}(F)<minH_{\delta}(G)$ and
(3) whenever $m\in\mathbb{N}$, $G_{1},\cdots,G_{m}\in P_{f}(\mathcal{T}_{0})$,
$G_{1}\subset G_{2}\subset\cdots\subset G_{m}$, and for each
$i\in\\{1,2,\cdots,m\\}$, $f_{i}\in G_{i}$, one has

$\sum_{i=1}^{m}\big{(}\alpha_{\delta}(G_{i})+\sum_{t\in
H_{\delta}(G_{i})}f_{i}(t)\big{)}\in A^{*}.$

Assume that $F=\\{f\\}$. Since $A^{*}$ is piecewise syndetic near zero, pick
by Theorem 3.4, for $\delta>0$, $a\in S\cap(0,\delta)$ and $L\in
P_{f}(\mathbb{N})$ such that $a+\sum_{t\in L}f(t)\in A^{*}$. Let
$\alpha_{\delta}(\\{f\\})=a$ and $H_{\delta}(\\{f\\})=L$.

Let $|F|>1$, $\alpha_{\delta}(G)$ and $H_{\delta}(G)$ have been defined for
all proper subsets $G$ of $F$ and for each $\delta>0$. Pick $\delta>0$, and
let

$K_{\delta}=\bigcup\\{H_{\delta}(G):G\mbox{ is a non-empty proper subset of
}F\\}$

and let $m=maxK_{\delta}$. Let

$\displaystyle M_{\delta}$ $\displaystyle=$
$\displaystyle\big{\\{}\sum_{i=1}^{n}\big{(}\alpha_{\delta}(G_{i})+\sum_{t\in
H_{\delta}(G_{i})}f_{i}(t)\big{)}:n\in\mathbb{N},\emptyset\neq
G_{1}\subset\cdots\subset G_{n}\subset F,$ $\displaystyle and$
$\displaystyle\\{f_{i}\\}_{i=1}^{n}\in\prod_{i=1}^{n}G_{i}\big{\\}}.$

Then $M_{\delta}$ is finite and by hypothesis (3), $M_{\delta}\subseteq
A^{*}$. Let $B=A^{*}\cap\bigcap_{x\in M_{\delta}}(-x+A^{*})$. Then $B\in p$ so
pick by Theorem 3.4 and Lemma 3.3, $a\in S\cap(0,\delta)$ and $L\in
P_{f}(\mathbb{N})$ such that $a+\sum_{t\in L}f(t)\in B$ for each $f\in F$. Let
$\alpha_{\delta}(F)=a$ and $H_{\delta}(F)=L$.

The hypothesis (1) is obvious. Since $minL>m$, we have the hypothesis (2) is
satisfied. To verify hypothesis (3), pick $\delta>0$ and $n\in\mathbb{N}$, let
$\emptyset\subset G_{1}\subset\cdots\subset G_{n}=F$, and let
$\\{f_{i}\\}_{i=1}^{n}\in\prod_{i=1}^{n}G_{i}$. If $n=1$, then
$\alpha(G_{1}\prod\delta)+\sum_{t\in H(G_{1})}f_{1}(t)=a+\sum_{t\in
L}f_{1}(t)\in B\subseteq A^{*}$. So assume that $n>1$ and let
$y=\sum_{i=1}^{n-1}\big{(}\alpha_{\delta}(G_{i})+\sum_{t\in
H_{\delta}(G_{i})}f_{i}(t)\big{)}$. Then $y\in M_{\delta}$ so $a+\sum_{t\in
L}f_{1}(t)\in B\subseteq(-y+A^{*})$ and thus
$\sum_{i=1}^{n}\big{(}\alpha_{\delta}(G_{i})+\sum_{t\in
H_{\delta}(G_{i})}f_{i}(t)\big{)}=y+a+\sum_{t\in L}f_{1}(t)\in A^{*}$ as
required. ∎

######  Definition 3.6.

Let $S$ be a dense subsemigroup of $\big{(}(0,\infty),+\big{)}$ and let
$A\subseteq S$. We say $A$ is a $C-$set near zero if and only if for each
$\delta\in(0,1)$, there exist functions
$\alpha_{\delta}:P_{f}(\mathcal{T}_{0})\rightarrow S$ and
$H_{\delta}:P_{f}(\mathcal{T}_{0})\rightarrow P_{f}(\mathbb{N})$ such that
(1) $\alpha_{\delta}(F)<\delta$ for each $F\in P_{f}(\mathcal{T}_{0})$,
(2) if $F,G\in P_{f}(\mathcal{T}_{0})$ and $F\subset G$, then
$maxH_{\delta}(F)<minH_{\delta}(G)$ and
(3) whenever $m\in\mathbb{N}$, $G_{1},\cdots,G_{m}\in P_{f}(\mathcal{T}_{0})$,
$G_{1}\subset G_{2}\subset\cdots\subset G_{m}$, and for each
$i\in\\{1,2,\cdots,m\\}$, $f_{i}\in G_{i}$, one has

$\sum_{i=1}^{m}\big{(}\alpha_{\delta}(G_{i})+\sum_{t\in
H_{\delta}(G_{i})}f_{i}(t)\big{)}\in A.$

Let $\Phi$ is the set of all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ for
which $f(n)\leq n$ for each $n\in\mathbb{N}$.

######  Theorem 3.7.

Let $S$ be a dense subsemigroup of $\big{(}(0,\infty),+\big{)}$ and let $A$ be
a $C$-set near zero in $S$, and for each $l\in\mathbb{N}$, let
$\\{y_{l,n}\\}_{n\in\mathbb{N}}\in\mathcal{T}_{0}$. There exist a sequence
$\\{a_{n}\\}_{n\in\mathbb{N}}$ in $S$ such that $a_{n}\rightarrow 0$ and a
sequence $\\{H_{n}\\}_{n\in\mathbb{N}}$ in $P_{f}(\mathbb{N})$ such that
$maxH_{n}<minH_{n+1}$ for each $n\in\mathbb{N}$ and such that for each
$f\in\Phi$

$FS\big{(}\\{a_{n}+\sum_{t\in
H_{n}}y_{f(n),t}\\}_{n\in\mathbb{N}}\big{)}\subseteq A.$

In particular, the above conclusion applies if $A$ is a central set near zero
in $S$.

###### Proof.

Pick $\alpha$ and $H$ as guaranteed by Definition 3.6. We may assume that the
sequences $\\{y_{l,n}\\}_{n\in\mathbb{N}}$ are distinct.

For $n\in\mathbb{N}$, let
$F_{n}=\\{\\{y_{l,t}\\}_{t\in\mathbb{N}},\\{y_{2,t}\\}_{t\in\mathbb{N}},\cdots,\\{y_{n,t}\\}_{t\in\mathbb{N}}\\}$
and let $a_{n}=\alpha_{\frac{1}{n}}(F_{n})<\frac{1}{n}$ and
$H_{n}=H_{\frac{1}{n}}(F_{n})$. Let $f\in\Phi$ be given. To see that

$FS\big{(}\\{a_{n}+\sum_{t\in
H_{n}}y_{f(n),t}\\}_{n\in\mathbb{N}}\big{)}\subseteq A,$

let $K\in P_{f}(\mathbb{N})$. Let $K=\\{n_{1},\cdots,n_{m}\\}$ where
$n_{1}<n_{2},\cdots<n_{m}$. Then $F_{n_{1}}\subset
F_{n_{2}}\subset\cdots\subset F_{n_{m}}$ and for each $i\in\\{1,\cdots,m\\}$,
$\\{y_{f(n_{i}),t}\\}_{t\in\mathbb{N}}\in F_{n_{i}}$ so

$\sum_{n\in K}(a_{n}+\sum_{t\in
H_{n}}y_{f(n),t})=\sum_{i=1}^{m}(\alpha_{\frac{1}{n_{i}}}(F_{n_{i}})+\sum_{t\in
H_{\frac{1}{n_{i}}}(F_{n_{i}})}y_{f(n_{i}),t})\in A.$

The ”in particular” is obvious. ∎

######  Definition 3.8.

Let $S$ be a dense subsemigroup of $\big{(}(0,\infty),+\big{)}$ and pick
$m\in\mathbb{N}$. We define

$\mathcal{V}_{m}=\\{\prod_{i=1}^{m}H_{i}\in P_{f}(\mathbb{N})^{m}:\mbox{ if
}m>1,\,1\leq t\leq m-1,\\\ \mbox{ then }maxH_{t}<minH_{t+1}\\},\\\ $
$\mathcal{J}_{m}=\\{\prod_{i=1}^{m}t(i)\in\mathbb{N}^{m}:t(1)<\cdots<t(m)\\},$

and

$S^{m}_{\delta}=S^{m}\cap\big{(}0,\frac{\delta}{m}\big{)}^{m}$

for $\delta>0$.

######  Definition 3.9.

Let $S$ be a dense subsemigroup of $\big{(}(0,\infty),+\big{)}$.
(a) Given $m\in\mathbb{N}$, $\delta>0$, $a\in S_{\delta}^{m+1}$,
$t\in\mathcal{J}_{m}$, and $f\in\mathcal{T}_{0}$, define

$x(m,a,t,f)=a(m+1)+\sum_{j=1}^{m}a(j)+f(t(j)).$

b) $J_{0}(S)=\\{p\in 0^{+}:\mbox{ for all }A\in p,A\mbox{ is a }J-\mbox{set
near zero}\\}.$

######  Lemma 3.10.

Let $S$ be a dense subsemigroup of $\big{(}(0,\infty),+\big{)}$ and let
$A\subseteq S$. Then $A$ is a $J$-set near zero if and only if for each $F\in
P_{f}(\mathcal{T}_{0})$ and for each $\delta>0$ there exist $m\in\mathbb{N}$,
$a\in S_{\delta}^{m+1}$, and $t\in\mathcal{J}_{m}$ such that for each $f\in
F$, $x(m,a,t,f)\in A$.

###### Proof.

The sufficiency of the statement for $J-$set is trivial. Assume that $A$ is a
$J-$set near zero. Pick $F\in P_{f}(^{\mathbb{N}}S)$ and $\delta>0$. Pick
$c\in\mathbb{N}$ and for $f\in F$, define $g_{f}\in\mathcal{T}_{0}$ by
$g_{f}(n)=f(n+c)$. Pick $b\in S$ and $H\in P_{f}(\mathbb{N})$ such that for
each $f\in F$, $b+\sum_{t\in H}g_{f}(t)\in A$. Let $m=|H|$, and let
$t=\big{(}t(1),\cdots,t(m)\big{)}$ enumerate $H$ in increasing order, let
$a(1)=b$ and for $j\in\\{2,\cdots,m+1\\}$, let $a(j)=c$. Then the proof is
complete. ∎

######  Lemma 3.11.

Let $S$ be a dense subsemigroup of $\big{(}(0,\infty),+\big{)}$, $A\subseteq
S$ is a $C-$set near zero if and only if for each $\delta>0$, there exist
$m_{\delta}:P_{f}(\mathcal{T}_{0})\rightarrow\mathbb{N}$,
$\alpha\in\prod_{F\in P_{f}(\mathcal{T}_{0})}S_{\delta}^{m_{\delta}(F)+1}$,
and $\tau\in\prod_{F\in P_{f}(\mathcal{T}_{0})}\mathcal{J}_{m_{\delta}(F)}$
such that
(1) if $F,G\in P_{f}(\mathcal{T}_{0})$ and $F\subset G$ then
$\tau(F)(m_{\delta}(F))<\tau(G)(1)$ for each $\delta>0$, and
(2) whenever $n\in\mathbb{N}$, $G_{1},\cdots,G_{n}\in P_{f}(\mathcal{T}_{0})$,
$G_{1}\subset G_{2}\subset\cdots\subset G_{n}$, and for each
$i\in\\{1,\cdots,n\\}$, $f_{i}\in G_{i}$, one has

$\sum_{i=1}^{n}x(m_{\delta}(G_{i}),\alpha(G_{i}),\tau(G_{i}),f_{i})\in A.$

###### Proof.

For the statement about $C-$sets the sufficiency is trivial. For the
necessity, pick $\alpha_{\delta}:P_{f}(\mathcal{T}_{0})\rightarrow S$ and
$H_{\delta}:P_{F}(\mathcal{T}_{0})\rightarrow P_{f}(\mathbb{N})$ for each
$\delta\in(0,1)$ as guaranteed by Definition of $C-$set near zero. Now pick
$c\in\mathbb{N}$ and for $f\in\mathcal{T}_{0}$ define
$g_{f}\in\mathcal{T}_{0}$ by $g_{f}(s)=f(s+c)$, for $s\in\mathbb{N}$. For
$F\in P_{f}(\mathcal{T}_{0})$ we define inductively on $|F|$ a set $K(F)\in
P_{f}(\mathcal{T}_{0})$ such that
(1) $\\{g_{f}:f\in F\\}\subseteq K(F)$ and
(2) if $\emptyset\neq G\subset F$, then $K(G)\subset K(F)$.

If $F=\\{f\\}$, let $K(F)=\\{g_{f}\\}$. Now let $|F|>1$ and $K(G)$ has been
defined for all proper nonempty subsets of $F$. Pick
$h\in\mathcal{T}_{0}\setminus\bigcup\\{K(G):\emptyset\neq G\subset F\\}$ and
let $K(F)=\\{h\\}\cup\\{g_{f}:f\in F\\}\cup\bigcup\\{K(G):\emptyset\neq
G\subset F\\}$.

Now for each $\delta\in(0,1)$, we define
$m_{\delta}:P_{f}(\mathcal{T}_{0})\rightarrow\mathbb{N}$,
$\alpha^{\prime}\in\prod_{F\in P_{f}(\mathcal{T}_{0})}S^{m_{\delta}(F)+1}$,
and $\tau\in\prod_{F\in P_{f}(\mathcal{T}_{0})}\mathcal{J}_{m_{\delta}(F)}$.

Let $F\in P_{f}(\mathcal{T}_{0})$ be given and let
$m_{\delta}(F)=|H_{\delta}(K(F))|$. Define $\alpha^{\prime}(F)\in
S^{m_{\delta}(F)+1}$ by, for $j\in\\{1,2,\cdots,m_{\delta}(F)+1\\}$,
$\alpha^{\prime}(F)(j)=\alpha_{\delta}(K(F))$ if $j=1$ and
$\alpha^{\prime}(F)(j)=c$ if $j>1$. Let
$\tau(F)=(\tau(F)(1),\cdots,\tau(F)(m_{\delta}(F))$ enumerate
$H_{\delta}(K(F))$. We need to show that
(1) if $F,G\in P_{f}(\mathcal{T}_{0})$ and $F\subset G$, then
$\tau(F)(m_{\delta}(F))<\tau(G)(1)$ for each $\delta\in(0,1)$, and
(2) whenever $n\in\mathbb{N}$, $G_{1},\cdots,G_{n}\in P_{f}(\mathcal{T}_{0})$,
$G_{1}\subset G_{2}\subset\cdots\subset G_{n}$, and for each
$i\in\\{1,\cdots,n\\}$, $f_{i}\in G_{i}$, one has

$\sum_{i=1}^{n}x(m_{\delta}(G_{i}),\alpha^{\prime}(G_{i}),\tau(G_{i}),f_{i})\in
A.$

To verify (1), let $F,G\in P_{f}(\mathcal{T}_{0})$ with $F\subset G$, then
$K(F)\subset K(F)$, and so
$\tau(F)(m_{\delta}(F))=maxH_{\delta}(K(F))<minH_{\delta}(K(G))=\tau(G)(1)$.

To verify (2), let $n\in\mathbb{N}$, $G_{1},\cdots,G_{n}\in
P_{f}(\mathcal{T}_{0})$, $G_{1}\subset G_{2}\subset\cdots\subset G_{n}$, and
for each $i\in\\{1,\cdots,n\\}$, let $f_{i}\in G_{i}$. Then $K(G_{1})\subset
K(G_{2})\subset\cdots\subset K(G_{n})$, and for each $f_{i}\in G_{i}$,
$g_{f_{i}}\in K(G_{i})$ so
$\sum^{n}_{i=1}(\alpha_{\delta}(K(G_{i})))+\sum_{t\in
H_{\delta}(K(G_{i}))}g_{f_{i}}(t))\in A$ and

$\displaystyle\sum^{n}_{i=1}$
$\displaystyle(\alpha_{\delta}(K(G_{i}))+\sum_{t\in
H_{\delta}(K(G_{i}))}g_{f_{i}}(t))$
$\displaystyle=\sum^{n}_{i=1}(\alpha_{\delta}(K(G_{i}))+\sum^{m_{\delta}(G_{i})}_{j=1}(f_{i}(\tau(G_{i})(j)+c))$
$\displaystyle=\sum^{n}_{i=1}(\sum^{m_{\delta}(G_{i})}_{j=1}(\alpha^{\prime}(G_{i})(j)+(f_{i}(\tau(G_{i})(j)))+\alpha^{\prime}(G_{i})(m_{\delta}(G_{i})+1))$
$\displaystyle=\sum^{n}_{i=1}x(m_{\delta}(G_{i}),\alpha^{\prime}(G_{i}),\tau(G_{i}),f_{i}).$

∎

######  Lemma 3.12.

Let $S$ be a dense subsemigroup of $\big{(}(0,\infty),+\big{)}$ and let
$A\subseteq S$. Let $A$ be a J-set near zero in $S$, then for each $F\in
P_{f}(\mathcal{T}_{0})$, each $\delta>0$ and each $n\in\mathbb{N}$, there
exist $m\in\mathbb{N}$, $a\in S_{\delta}^{m+1}$, and $y\in\mathcal{J}_{m}$
such that $y(1)>n$ and for each $f\in F$, $x(m,a,y,f)\in A$.

###### Proof.

Pick $F\in P_{f}(\mathcal{T}_{0})$, $\delta>0$ and $n\in\mathbb{N}$. For each
$f\in F$ define $g_{f}\in\mathcal{T}_{0}$ by, for $u\in\mathbb{N}$,
$g_{f}(u)=f(u+n)$. Pick $m\in\mathbb{N}$, $a\in S_{\delta}^{m+1}$ and
$t\in\mathcal{J}_{m}$ such that for each $f\in F$, $x(m,a,t,g_{f})\in A$.
Define $y\in\mathcal{J}_{m}$ by $y(i)=n+t(i)$ for $i\in\\{1,2,\cdots,m\\}$.
Then $y(1)>1$ and for each $f\in F$, $x(m,a,y,f)\in A$. ∎

######  Theorem 3.13.

Let $S$ be a dense subsemigroup of $\big{(}(0,\infty),+\big{)}$. Then
$J_{0}(S)$ is a compact two sided ideal of $\beta S$.

###### Proof.

Trivially $J_{0}(S)$ is topologically closed in $\beta S$. Let $p\in J_{0}(S)$
and let $q\in\beta S$. We show $q+p\in J_{0}(S)$ and $p+q\in J_{0}(S)$.
To see $q+p\in J_{0}(S)$, let $A\in q+p$ and let $F\in
P_{f}(\mathcal{T}_{0})$. Then $\\{b\in S:-b+A\in p\\}\in q$ so pick $b\in S$
such that $-b+A\in p$. Pick $m\in\mathbb{N}$, $\delta\in(0,1)$, $a\in
S_{\delta}^{m+1}$, and $t\in\mathcal{J}_{m}$ such that for $f\in F$,
$x(m,a,t,f)\in-b+A$. Define $c\in S_{\delta}^{m+1}$ by $c(1)=b+a(1)$ and
$c(j)=a(j)$ for $j\in\\{2,3,...,m+1\\}$. Then for each $f\in F$,
$x(m,c,t,f)\in A$.

To see $p+q\in J_{0}(S)$, let $A\in p+q$ and let $B=\\{x\in S:-x+A\in q\\}$.
Then $B\in p$ so for $F\in P_{f}(\mathcal{T}_{0})$ and $\delta\in(0,1)$, pick
$m\in\mathbb{N}$, $a\in S_{\delta}^{m+1}$, and $t\in\mathcal{J}_{m}$ such that
for $f\in F$, $x(m,a,t,f)\in B$. Then $\bigcap_{f\in F}(-x(m,a,t,f)+A)\in q$
so pick $b\in\bigcap_{f\in F}-x(m,a,t,f)+A$. Define $c\in S_{\delta}^{m+1}$ by
$c(m+1)=a(m+1)+b$ and $c(j)=a(j)$ for $j\in\\{1,2,...,m\\}$. Then for $f\in
F$, $x(m,c,t,f)\in A$. ∎

######  Lemma 3.14.

Let $S$ be a dense subsemigroup of $\big{(}(0,\infty),+\big{)}$. Pick
$\delta\in(0,1)$, and let $m,r\in\mathbb{N}$, let $a\in S_{\delta}^{m+1}$, let
$t\in\mathcal{J}_{m}$, and for each $y\in\mathbb{N}$, let $c_{y}\in
S_{\delta}^{r+1}$ and $z_{y}\in\mathcal{J}_{r}$ be a such that for each
$y\in\mathbb{N}$, $z_{y}(r)<z_{y+1}(1)$. Then there exist $u\in\mathbb{N}$,
$d\in S_{\delta}^{u+1}$, and $q\in\mathcal{J}_{u}$ such that for each
$f\in\mathcal{T}_{0}$,

$(\sum_{j=1}^{m}a(j)+x(r,c_{t}(j),z_{t}(j),f))+a(m+1)=x(u,d,q,f).$

###### Proof.

We have

$\sum_{j=1}^{m}(a(j)+x(r,c_{t(j)},z_{t(j)},f))+a(m+1)=\\\
\sum_{j=1}^{m}\big{(}a(j)+c_{t(j)}(r+1)+\sum_{k=1}^{r}(c_{t(j)}(k)+f(z_{t(j)}(k)))\big{)}+a(m+1)=\\\
\sum_{j=1}^{m}a(j)+\sum_{j=1}^{m}c_{t(j)}(r+1)+\sum_{j=1}^{m}\sum_{k=1}^{r}(c_{t(j)}(k)+f(z_{t(j)}(k))+a(m+1).\\\
$

Now let $u=m\cdot r$. For $j\in\\{1,2,...,m\\}$ and $p\in\\{1,2,...,r\\}$, let
$q((j-1)\cdot r+p)=z_{t(j)}(p)$. Let $d(1)=a(1)+c_{t(1)}(1)$, let
$d(u+1)=c_{t(m)}(r+1)+a(m+1)$, for $j\in\\{1,2,...,m-1\\}$, let $d(j\cdot
r+1)=c_{t(j)}(r+1)+a(j+1)+c_{t(j+1)}(1)$, and for $j\in\\{1,2,...,m\\}$ and
$p\in\\{2,3,...,r\\}$, let $d((j-1)\cdot r+p)=c_{t(j)}(p)$. So this complete
the proof. ∎

######  Lemma 3.15.

Let $S$ be a dense subsemigroup of $\big{(}(0,\infty),+\big{)}$, and let
$A_{1}$ and $A_{2}$ be subsets of $S$. If $A_{1}\cup A_{2}$ is a $J$-set near
zero, then either $A_{1}$ is a J-set near zero or $A_{2}$ is a J-set neat
zero.

###### Proof.

Suppose not and pick $F_{1}$ and $F_{2}$ in $P_{f}(^{\mathbb{N}}S)$ and
$\delta>0$ such that for each $i\in\\{1,2\\}$, each $u\in\mathbb{N}$, each
$d\in S_{\delta}^{u+1}$, and each $q\in\mathcal{J}_{u}$, there is some $f\in
F_{i}$ such that $x(u,d,q,f)\notin A_{i}$.

Let $F=F_{1}\cup F_{2}$, $k=\mid F\mid$, and write
$F=\\{f_{1},f_{2},...,f_{k}\\}$. Pick by Lemma, some $n\in\mathbb{N}$ such
that whenever length $n$ words over the alphabet $\\{1,2,...,k\\}$ are
2-colored, there is a variable word $w(v)$ beginning and ending with a
constant and without successive occurrences of $v$ such that
$\\{w(l):l\in\\{1,2,...,k\\}\\}$ is monochromatic.

Let $W$ be the set of length $n$ words over $\\{1,2,...,k\\}$. For
$w=b_{1},b_{2},...,b_{n}\\\ \in W$(where each $b_{i}\in\\{1,2,...,k\\}$),
define $g_{w}:\mathbb{N}\rightarrow S$ by, $y\in S$,
$g_{w}(y)=\sum^{n}_{i=1}f_{b_{i}}(ny+ix)$ where $x\in S$. Since $A$ is a
$J_{S}$-set near zero, pick $m\in\mathbb{N}$, $a\in S_{\delta}^{m+1}$,
$t\in\mathcal{J}_{m}$ such that for all $w\in W$, $x(m,a,t,g_{w})\in A$.
Define $\varphi:W\rightarrow\\{1,2\\}$ by $\varphi(w)=1$ if $x(m,a,t,g_{w})\in
A_{1}$ and $\varphi(w)=2$ otherwise. Pick a variable word $w(v)$, beginning
and ending with a constant and without successive occurrences of $v$ such that
$\varphi$ is constant on $\\{w(l):\l\in\\{1,2,...,k\\}\\}$. Assume without
loss of generality that $\varphi(w(l))=1$ for all $j\in\\{1,2,...,k\\}$. That
is, for all $l\in\\{1,2,...,k\\}$,

$(\sum^{m}_{j=1}a(j)+g_{w(l)}(t(j)))+a(m+1)=x(m,a,t,g_{w(l)})\in A_{1}.$

Let $w(v)=b_{1}b_{2}\cdots b_{n}$ where
each$b_{i}\in\\{1,2,,...,k\\}\cup\\{v\\}$, some $b_{i}=v$, $b_{1}\neq v$,
$b_{n}\neq v$, and if $b_{i}=v$, then $b_{i+1}\neq v$. Let $r$ be the number
of occurrences of $v$ in $w(v)$ and pick $L\in\mathcal{V}_{r+1}$ and
$s\in\mathcal{J}_{r}$ such that for each $p\in\\{1,\cdots,r\\}$,
$maxL_{p}<s(p)<minL_{p+1}$,

$\bigcup_{p=1}^{r+1}L_{p}=\\{i\in\\{1,\cdots,n\\}:b_{i}\in\\{1,\cdots,k\\}\\}$

and $\\{s(1),\cdots,s(r)\\}=\\{i\in\\{1,\cdots,n\\}:b_{i}=v\\}$. (For example,
if $w(v)=12v131v2v1121v32$, then $r=4$,
$L=(\\{1,2\\},\\{4,5,6\\},\\{8\\},\\{10,11,12\\},\\{14,15\\})$, and
$s=(3,7,9,14)$.)

We shall show now that, given $y\in\mathbb{N}$, there exist $c_{y}\in
S_{\delta}^{r+1}$ and $z_{y}\in\mathcal{J}_{r}$ such that for all
$l\in\\{1,2,...,k\\}$, $g_{w(l)}(y)=x(r,c_{y},z_{y},f_{l})$ and further, for
each $y$, $z_{y}(r)<z_{y+1}(1)$. So let $y\in\mathbb{N}$ be given. For
$p\in\\{1,2,\cdots,r+1\\}$, let $c_{y}(p)=\sum_{i\in L(p)}f_{b_{i}}(ny+i)$ and
for $p\in\\{1,2,\cdots,r\\}$, let $z_{y}(p)=ny+s(p)$. To see that these are as
required, first note that $z_{y}(r)\leq ny+n<z_{y+1}(1)$. Now let
$l\in\\{1,\cdots,k\\}$ be given. Then $w(l)=d_{1}d_{2}\cdots d_{n}$ where for
$i\in\\{1,2,\cdots,n\\}$, $d_{i}=b_{i}$ if $i\in\bigcup^{r+1}_{p=1}L_{p}$ and
$d_{i}=l$ if $i\in\\{s(1),s(2),\cdots,s(r)\\}$.

Therefore

$\displaystyle g_{w(l)}(y)$ $\displaystyle=\sum^{n}_{i=1}f_{d_{i}}(ny+i)$
$\displaystyle=(\sum^{r}_{p=1}(\sum_{i\in
L(p)}f_{b_{i}}(ny+i))+f_{l}(ny+s(p)))+\sum_{i\in L(r+1)}f_{b_{i}}(ny+i)$
$\displaystyle=(\sum^{r}_{p=1}c_{y}(p)+f_{l}(z_{y}(p)))+c_{y}(r+1)$
$\displaystyle=x(r,c_{y},z_{y},f_{l})$

as required.

Now pick Lemma 3.14, $u\in\mathbb{N}$, $d\in S_{\delta}^{u+1}$,
$q\in\mathcal{J}_{u}$ such that for each $f\in^{\mathbb{N}}S$,

$(\sum^{m}_{j=1}a(j)+x(r,c_{t(j)},z_{t(j)},f))+a(m+1)=x(u,d,q,f).$

Pick $l\in\\{1,2,\cdots,k\\}$ such that $f_{l}\in F_{1}$ and
$x(u,d,q,f_{l})\notin A_{1}$. But

$\displaystyle x(u,d,q,f_{l})$
$\displaystyle=(\sum^{m}_{j=1}a(j)+x(r,c_{t(j)},z_{t(j)},f_{l}))+a(m+1)$
$\displaystyle=(\sum^{m}_{j=1}a(j)+g_{w(l)}(t(j)))+a(m+1)$
$\displaystyle=x(m,a,t,g_{w(l)})\in A_{1},$

a contradiction. ∎

######  Theorem 3.16.

Let $S$ be a dense subsemigroup of $\big{(}(0,\infty),+\big{)}$, let
$A\subseteq S$. Then $\overline{A}\cap J_{0}(S)\neq\emptyset$ if and only if
$A$ is a $J_{S}$-set near zero.

###### Proof.

The necessity is trivial. By Lemma 3.15, $J_{S}$-sets are partition regular.
So, if $A$ is a $J_{S}$-set near zero, by Theorem 1.2, there is some
$p\in\beta S$ such that $A\in p$ and for every $B\in p$, $B$ is a $J_{S}$-set
near zero. ∎

######  Corollary 3.17.

Let $S$ be a dense subsemigroup of $\big{(}(0,\infty),+\big{)}$, and let $A$
be a piecewise syndetic near zero subset of $S$. Then $A$ is a $J_{S}$-set
near zero.

###### Proof.

By Theorem 2.4, $\overline{A}\cap K(0^{+}(S))\neq\emptyset$. Since
$K(0^{+}(S))\subseteq J_{0}(S)$, so $\overline{A}\cap J_{0}(S)\neq\emptyset$
so by Theorem 3.16, $A$ is a $J_{S}$-set near zero. ∎

######  Theorem 3.18.

Let $S$ be a semigroup and let $A\subseteq S$. If there is an idempotent in
$\overline{A}\cap J_{0}(S)$, then $A$ is a C-set near zero.

###### Proof.

Pick $p=p+p\in\overline{A}\cap J_{0}(S)$. Recall that $A^{*}=\\{x\in A:-x+A\in
p\\}$ and, by lemma 1.1, if $x\in A^{*}$, then $-x+A^{*}\in p$. For every
$\delta\in(0,1)$ we define $m_{\delta}(F)$ and $\alpha(F)$ and $\tau(F)$ for
$F\in P_{f}(\mathcal{T}_{0})$ by induction on $|F|$ so that

(1) if $F,G\in P_{f}(\mathcal{T}_{0})$ and $F\subset G$ then
$\tau(F)(m_{\delta}(F))<\tau(G)(1)$ for each $\delta>0$, and

(2) whenever $n\in\mathbb{N}$, $G_{1},\cdots,G_{n}\in P_{f}(\mathcal{T}_{0})$,
$G_{1}\subset G_{2}\subset\cdots\subset G_{n}$, and for each
$i\in\\{1,\cdots,n\\}$, $f_{i}\in G_{i}$, one has

$\sum_{i=1}^{n}x(m_{\delta}(G_{i}),\alpha(G_{i}),\tau(G_{i}),f_{i})\in A.$

Assume first that $F=\\{f\\}$. Then $A^{*}$ is a $J_{S}$-set near zero so pick
$m_{\delta}(F)\in\mathbb{N}$, $\alpha(F)\in S^{m_{\delta}(F)+1}$, and
$\tau(F)\in\mathcal{J}_{m_{\delta}(F)}$ such that

$x(m_{\delta}(F),\alpha(F),\tau(F),f)\in A^{*}.$

Now assume that $|F|>1$ and that $m_{\delta}(G)$, $\alpha(G)$, and $\tau(G)$
have been defined for all non-empty proper subsets $G$ of $F$ and for each
$\delta>0$. Pick $\delta>0$, and let
$k=max\\{\tau(G)(m_{\delta}(G)):\emptyset\neq G\subsetneq F\\}$. Let

$M_{\delta}=\\{\sum_{i=1}^{m}x(m_{\delta}(G_{i}),\alpha(G_{i}),\tau(G_{i}),f_{i}):n\in\mathbb{N},\\\
\emptyset\neq G_{1}\subset G_{2}\subset\cdots\subset
G_{n}=F,and\,\,\\{f_{i}\\}_{i=1}^{n}\in\Pi_{i=1}^{n}G_{i}\\}.\\\ $

Let $B=A^{*}\cap\bigcap_{b\in M_{\delta}}(-b+A^{*})$. Since $M_{\delta}$ is a
finite subset of $A^{*}$, $B\in p$ and therefor $B$ is a J-set near zero. Pick
by Lemma 3.10, $m_{\delta}(F)\in\mathbb{N}$, $\alpha(F)\in
S^{m_{\delta}(F)+1}$ , and $\tau(F)\in\mathcal{J}_{m_{\delta}(F)}$ such that
$\tau(F)(1)>k$ and for each $f\in F$, $x(m_{\delta}(F),\alpha(F),\tau(F),f)\in
B$.

Hypothesis $(1)$ is satisfied directly. To verify hypothesis $(2)$, let
$n\in\mathbb{N}$, let $\emptyset\neq G_{1}\subset G_{2}\subset\cdots\subset
G_{n}=F$, and for each $i\in\\{1,2,\cdots,n\\}$, let $f_{i}\in G_{i}$. If
$n=1$, then $x(m_{\delta}(G_{1}),\alpha(G_{1}),\tau(G_{1}),f_{i})\in
B\subseteq A^{*}$, so assume that $n>1$. Let
$b=\sum_{i=1}^{n-1}x(m_{\delta}(G_{i}),\alpha(G_{i}),\tau(G_{i}),f_{i})$ then
$b\in M_{\delta}$ so $x(m_{\delta}(G_{n}),\alpha(G_{n}),\tau(G_{n}),f_{i})\in
B\subseteq-b+A^{*}$ so

$\sum_{i=1}^{n-1}x(m_{\delta}(G_{i}),\alpha(G_{i}),\tau(G_{i}),f_{i})\in
A^{*}$

as required. ∎

######  Corollary 3.19.

Let $S$ be a dense subsemigroup of $((0,\infty),+)$ and let $A$ be a central
set near zero in $S$. Then $A$ is a $C$-set near zero.

###### Proof.

It is obvious. ∎

######  Theorem 3.20.

Let $S$ be a dense subsemigroup of $((0,\infty),+)$, let $A$ be a central
subset near zero of $S$, and for each $l\in\mathbb{N}$, let
$\\{y_{l,n}\\}_{n\in\mathbb{N}}$ be a sequences in $S$ such that
$lim_{n\rightarrow\infty}y_{l,n}=0$ for each $l\in\mathbb{N}$. Given
$l,m\in\mathbb{N}$, $a\in S^{m+1}_{\frac{1}{m}}$, and $H\in\mathcal{V}_{m}$,
let

$w(a,H,l)=\big{(}\sum_{i=1}^{m}\big{(}a(i)+\sum_{t\in
H(i)}y_{l,t}\big{)}\big{)}+a(m+1).$

There exist sequences $\\{m(n)\\}_{n\in\mathbb{N}}$,
$\\{a_{n}\\}_{n\in\mathbb{N}}$, and $\\{H_{n}\\}_{n\in\mathbb{N}}$ such that

(1) for each $n\in\mathbb{N}$, $m(n)\in\mathbb{N}$, $a_{n}\in
S_{\frac{1}{m(n)}}^{m(n)+1}$, $H_{n}\in\mathcal{V}_{m(n)}$, and
$maxH_{n,m(n)}<minH_{n+1,1}$, and

(2) for each $f\in\Phi$,
$FS\big{(}\\{w(a_{n},H_{n},f(n))\\}_{n\in\mathbb{N}}\big{)}\subseteq A$.

###### Proof.

As in the proof of Theorem 3.7, we may assume that the sequences
$\\{y_{l,n}\\}_{n\in\mathbb{N}}$ are all distinct. $A$ is central near zero,
so $A$ is a $C$-set. For each $k\in\mathbb{N}$, pick
$m_{\frac{1}{k}}^{\prime}:P_{f}(\mathcal{T}_{0})\rightarrow\mathbb{N}$,
$\alpha\in\prod_{F\in
P_{f}(\mathcal{T}_{0})}S_{\frac{1}{k}}^{m_{\frac{1}{k}}^{\prime}(F)+1}$, and
$\tau\in\prod_{F\in P_{f}(\mathcal{T}_{0})}\mathcal{J}_{m_{k}^{\prime}(F)}$ as
guaranteed by the fact that $A$ is a $C$-set. For each $n\in\mathbb{N}$, let
$F_{n}=\\{\\{y_{1,t}\\}_{t\in\mathbb{N}},\cdots,\\{y_{n,t}\\}_{t\in\mathbb{N}}\\}$,
$m(n)=m^{\prime}_{\frac{1}{n}}(F_{n})$, $a_{n}=\alpha(F_{N})$ and

$H_{n}=\big{(}\\{\tau(F_{n})(1)\\},\cdots,\\{\tau(F_{n})(m(n))\\}\big{)}.$

To see that $m(n)$, $a_{n}$ and $H_{n}$ are as required, let $f\in\Phi$ and
let $K\in P_{f}(\mathbb{N})$. Enumerate $K$ in order as
$n(1),n(2),\cdots,n(l)$. For $i\in\\{1,\cdots,l\\}$, let
$g_{i}=\\{y_{f(n(i)),t}\\}_{t\in\mathbb{N}}$. Then

$\sum_{n\in
K}w(a_{n},H_{n},f(n))=\sum_{i=1}^{l}x\big{(}m^{\prime}_{\frac{1}{n(i)}}(F_{n(i)}),\alpha(F_{n(i)}),\tau(F_{n(i)}),g_{i}\big{)}\in
A.$

∎

######  Lemma 3.21.

Let $J$ be a set, let $(D,\leq)$ be a directed set, and let $S$ be a dense
subsemigroup of $((0,\infty),+)$. Let $\\{T_{i}\\}_{i\in D}$ be a decreasing
family of nonempty subsets of $S$ such that

1) $0\in cl_{\mathbb{R}}T_{i}$,

2) $\bigcap_{i\in D}T_{i}=\emptyset$, and

3) for each $i\in D$ and each $x\in T_{i}$ there is some $j\in D$ such that
$x+T_{j}\subseteq T_{i}$.

Let $Q=\bigcap_{i\in D}cl_{\beta S_{d}}T_{i}$. Then $Q$ is a compact
subsemigroup of $0^{+}(S)$. Let $\\{E_{i}\\}_{i\in D}$ and $\\{I_{i}\\}_{i\in
D}$ be decreasing families of nonempty subsets of $\Pi_{t\in J}S$ with the
following properties:

$(a)$ for each $i\in D$, $I_{i}\subseteq E_{i}\subseteq\Pi_{t\in J}T_{i}$,

$(b)$ for each $i\in D$ and each $\overrightarrow{x}\in I_{i}$ there exists
$j\in D$ such that $\overrightarrow{x}+E_{j}\subseteq I_{i}$, and

$(c)$ for each $i\in D$ and each $\overrightarrow{x}\in E_{i}\setminus I_{i}$
there exists $j\in D$ such that $\overrightarrow{x}+E_{j}\subseteq E_{i}$ and
$\overrightarrow{x}+I_{j}\subseteq I_{i}$.

Let $Y=\Pi_{t\in J}0^{+}(S)$, let $E=\bigcap_{i\in D}cl_{Y}E_{i}$, and let
$I=\bigcap_{i\in D}cl_{Y}I_{i}$. Then $E$ is a subsemigroup of $\Pi_{t\in J}Q$
and $I$ is an ideal of $E$. If, in addition, either

$(d)$ for each $i\in D$, $T_{i}=S$ and $\\{a\in S:\overline{a}\notin E_{i}\\}$
is not piecewise syndetic near zero, or

$(e)$ for each $i\in D$ and each $a\in T_{i}$ , $\overline{a}\in E_{i}$,

then given any $p\in K(Q)$, one has $\overline{p}\in E\cap K(\Pi_{t\in
J}Q)=K(E)\subseteq I$.

###### Proof.

By Theorem 4.20 in [7], $Q$ is a subsemigroup of $0^{+}(S)$. For the proof
that $E$ is a subsemigroup of $\Pi_{t\in J}Q$ and $I$ is an ideal of $E$, see
the proof of Lemma 14.9 in [7].

To complete the proof, assume that $(d)$ or $(e)$ holds. It suffices to
establish

$\mbox{if }p\in K(Q),\mbox{ then }\overline{p}\in E.\,\,\,\,\,(*)$

Indeed, assume we have established $(*)$. Then $\overline{p}\in E\cap\Pi_{t\in
J}K(Q)$ and $\Pi_{t\in J}K(Q)=K(\Pi_{t\in J}Q)$ by Theorem 2.23 in [7]. Then
by Theorem 1.65 in [7], $K(E)=E\cap K(\Pi_{t\in J}Q)$ and, since $I$ is an
ideal of $E$, $K(E)\subseteq I$.

To establish $(*)$, let $p\in K(Q)$ be given. To see that $\overline{p}\in E$,
let $i\in D$ be given and let $U$ be a neighborhood of $\overline{p}$. Pick
$F\in P_{f}(J)$ and for each $t\in F$ pick some $A_{t}\in p$ such that
$\bigcap_{t\in F}\pi_{t}^{-1}[cl_{\beta S_{d}}A_{t}]\subseteq U$, where
$\pi_{t}$ is projection for $t\in J$.

Assume now that $(d)$ holds. Since $p\in K(0^{+}(S)$ and $\\{a\in
S:\overline{a}\notin E_{i}\\}$ is not piecewise syndetic near zero, so by
Theorem 2.4, $\\{a\in S:\overline{a}\notin E_{i}\\}\notin p$ and hence
$\\{a\in S:\overline{a}\in E_{i}\\}\in p$. Now pick $a\in\big{(}\bigcap_{t\in
F}A_{t}\big{)}\cap\\{a\in S:\overline{a}\in E_{i}\\}$. Then $\overline{a}\in
U\cap E_{i}$.

If $(e)$ holds, see the proof of Lemma 14.9 in [7]. ∎

######  Theorem 3.22.

Let $S$ be a dense subsemigroup of $((0,\infty),+)$ and let $A\subseteq S$.
Then $A$ is a C-set near zero if and only if there is an idempotent in
$\overline{A}\cap J_{0}(S)$.

###### Proof.

The sufficiency is obvious.

Pick for every $k\in\mathbb{N}$, there exist
$m_{\frac{1}{k}}:P_{f}(\mathcal{T}_{0})\rightarrow\mathbb{N}$,
$\alpha\in\Pi_{F\in P_{f}(\mathcal{T}_{0})}S^{m_{\frac{1}{k}}(F)+1}$ , and
$\tau\in\Pi_{F\in P_{f}(\mathcal{T}_{0})}\mathcal{J}_{m_{\frac{1}{k}}(F)}$ as
for each $\frac{1}{k}\in(0,1]$ guaranteed by the fact that $A$ is a C-set near
zero. For $F\in P_{f}(\mathcal{T}_{0})$ and $k\in\mathbb{N}$ define

$T_{F,k}=\\{\sum_{i=1}^{n}x(m_{\frac{1}{k}}(F_{i}),\alpha(F_{i}),\tau(F_{i}),f_{i}):n\in\mathbb{N},\forall
F_{i}\in P_{f}(\mathcal{T}_{0}),\\\ F_{1}\subset F_{2}\subset\cdots\subset
F_{n},\mbox{ and for each }\\\ i\in\\{1,2,...,\\},f_{i}\in F_{i}\\}.\\\ $

It is obvious that if $F,G\in P_{f}(\mathcal{T}_{0})$, then $T_{F\cup
G,k}\subseteq T_{F,k}\cap T_{G,k}$ for each $k\in\mathbb{N}$, so
$Q_{k}=\bigcap_{F\in P_{f}(\mathcal{T}_{0})}\overline{T_{F,k}}$ is a non-empty
set for each $k\in\mathbb{N}$. Pick $k\in\mathbb{N}$, we show $Q_{k}$ is a
subsemigroup of $0^{+}(S)$.

For this it suffices by Theorem 4.20 in [7], we show that for all $F\in
P_{f}(\mathcal{T}_{0})$ and all $u\in T_{F,k}$, there is some $G\in
P_{f}(\mathcal{T}_{0})$ such that $u+T_{G,k}\subseteq T_{F,k}$. So let $F\in
P_{f}(\mathcal{T}_{0})$ and $u\in T_{F}$ be given. Pick $n\in\mathbb{N}$,
strictly increasing $\\{F_{i}\\}_{i=1}^{n}$ in $P_{f}(\mathcal{T}_{0})$ such
that $F\subset F_{1}$, and $f\in\Pi_{i=1}^{n}F_{i}$ such that

$u=\sum_{i=1}^{n}x(m_{\delta}(F_{i}),\alpha(F_{i}),H_{\delta}(F_{i}),f_{i}).$

Then $u+T_{F_{n},k}\subseteq T_{F,k}$. Therefore for each $k\in\mathbb{N}$,
$Q_{k}$ is a compact subsemigroup of $0^{+}(S)$.

Now for some $k\in\mathbb{N}$, we show that $K(Q_{k})\subseteq\overline{A}\cap
J_{0}(S)$ so that any idempotent in $K(Q)$ establishes the theorem. We have
that each $T_{F,k}\subseteq\overline{A}$ so $Q_{k}\subseteq\overline{A}$. Let
$p\in K(Q_{k})$. We need to show that $p\in J_{0}(S)$, so let $B\in p$. We
shall show that $B$ is a $J$-set near zero. So let $F\in
P_{f}(\mathcal{T}_{0})$. We shall produce $v\in\mathbb{N}$ $c\in
S_{\frac{1}{k}}^{v+1}$, and $t\in\mathcal{J}_{v}$ such that for each $f\in F$,
$x(v,c,t,f)\in B$.

We apply Lemma 3.21 with $J=F$ and $D=\\{G\in
P_{f}(\mathcal{T}_{0}):F\subseteq G\\}$. Pick $k\in\mathbb{N}$, and note that
$Q_{k}=\bigcap_{G\in D}\overline{T_{G,k}}$ as in Lemma 3.21. For $G\in D$ we
shall define a subset $I_{G}$ of $\Pi_{f\in F}T_{G,k}$ as follows. Let
$w\in\Pi_{f\in F}T_{G,k}$ then $w\in I_{G}$ if and only if there is some
$n\in\mathbb{N}-\\{1\\}$ such that there exist $C_{1}$, $C_{2}$,
$\\{G_{i}\\}_{i=1}^{n}$ and $\eta$ such that

$(1)$ $C_{1}$ and $C_{2}$ are disjoint nonempty sets and
$\\{1,2,...,n\\}=C_{1}\cup C_{2}$,

$(2)$ $\\{G_{i}\\}_{i=1}^{n}$ is strictly increasing in
$P_{f}(\mathcal{T}_{0})$ with $G\subset G_{1}$, and

$(3)$ $\eta\in\Pi_{i\in C_{1}}G_{i}$ and for each $f\in F$, if
$\gamma_{f}\in\Pi_{i=1}^{n}G_{i}$ is defined by

$\gamma_{f}(i)=\left\\{\begin{array}[]{ll}\eta_{i}&\text{if }i\in C_{1}\\\
f&\text{if }i\in C_{2}\end{array}\right.$

then
$w(f)=\sum_{i=1}^{n}x(m_{\frac{1}{k}}(G_{i}),\alpha(G_{i}),\tau(G_{i}),\gamma_{f}(i)$.

For $G\in D$, note that $I_{G}\neq\emptyset$ and let
$E_{G}=I_{G}\bigcup\\{\overline{b}:b\in T_{G}\\}$.

We claim that $\\{E_{G}\\}_{G\in D}$ and $\\{I_{G}\\}_{G\in D}$ satisfy
statements $(a),(b),(c)$ and $(e)$ of Lemma 3.21. Statements $(a)$ and $(e)$
hold trivially.

To verify $(b)$, let $G\in D$ and $w\in I_{G}$. Pick
$n,C_{1},C_{2},\\{G_{i}\\}_{i=1}^{n}$ and $\eta$ as guaranteed by the fact
that $w\in I_{G}$. We claim that $w+E_{G_{n}}\subseteq I_{G}$. So let $z\in
E_{G_{n}}$.

Assume first that $z=\overline{b}$ for some $b\in T_{G_{n}}$. Pick
$n^{\prime}\in\mathbb{N}$, strictly increasing
$\\{F_{i}\\}_{i=1}^{n^{\prime}}$ in $P_{f}(\mathcal{T}_{0})$ with
$G_{n}\subset F_{1}$, and $\eta^{\prime}\in\Pi_{i=1}^{n^{\prime}}F_{i}$ such
that

$b=\sum_{i=1}^{n}x(m_{\delta}(F_{i}),\alpha(F_{i}),\tau_{(}F_{i}),\tau(F_{i}),\eta^{\prime}(i)).$

Let $C_{1}^{\prime\prime}=C_{1}\cup\\{n+1,n+2,...,n+n^{\prime}\\}$ and for
$i\in\\{1,2,...,n+n^{\prime}\\}$ and

$L_{i}=\left\\{\begin{array}[]{ll}G_{i}&\text{if }i\leq n\\\ F_{i-n}&\text{if
}i>n.\end{array}\right.$

Define $\eta^{\prime\prime}\in\Pi_{i\in G_{1}^{\prime\prime}}L_{i}$ by, for
$i\in C_{1}^{\prime\prime}$,

$\eta^{\prime\prime}(i)=\left\\{\begin{array}[]{ll}\eta(i)&\text{if }i\leq
n\\\ \eta^{\prime}(i-n)&\text{if }i>n.\end{array}\right.$

Then $n+n^{\prime}$ ,
$C_{1}^{\prime\prime},C_{2}^{\prime\prime},\\{G_{i}\\}_{i=1}^{n+n^{\prime}}$,
and $\eta^{\prime\prime}$ establish that $w+z\in I_{G}$.

Now assume that $z\in I_{G_{n}}$. Pick
$n^{\prime},C_{1}^{\prime},C_{2}^{\prime},\\{F_{i}\\}_{i=1}^{n^{\prime}}$ and
$\eta^{\prime}$ as guaranteed by the fact that $z\in I_{G_{n}}$. Let
$C_{1}^{\prime\prime}=C_{1}\cup\\{n+i:i\in C_{1}^{\prime}\\}$, let
$C_{2}^{\prime\prime}=C_{2}\cup\\{n+i:i\in C_{2}^{\prime}\\}$, and for
$i\in\\{1,2,\cdots,n+n^{\prime}\\}$ let

$L_{i}=\left\\{\begin{array}[]{ll}G_{i}&\text{if }i\leq n\\\ F_{i-n}&\text{if
}i>n.\end{array}\right.$

Define $\eta^{\prime\prime}\in\Pi_{i\in C_{1}^{\prime\prime}}L_{i}$ by, for
$i\in C_{1}^{\prime\prime}$,

$\eta^{\prime\prime}(i)=\left\\{\begin{array}[]{ll}\eta(i)&\text{if }i\leq
n\\\ \eta^{\prime}(i-n)&\text{if }i>n.\end{array}\right.$

Then
$n+n^{\prime},C_{1}^{\prime\prime},C_{2}^{\prime\prime},\\{L_{i}\\}_{i=1}^{n+n^{\prime}\prime}$,
and $\eta^{\prime\prime}$ establish that $w+z\in I_{G}$.

To verify $(c)$ let $G\in D$ and let $w\in E_{G}\setminus I_{G}$. pick $b\in
T_{G}$ such that $w=\bar{b}$. Pick $n\in\mathbb{N}$, strictly increasing
$\\{G_{i}\\}_{i=1}^{n}$ in $P_{f}(\mathcal{T}_{0})$ with $G\subset G_{1}$, and
$\eta\in\Pi_{i=1}^{n}G_{i}$ such that
$b=\sum_{i=1}^{n}x(m_{\frac{1}{k}}(G_{i}),\alpha(G_{i}),\tau(G_{i}),\eta(i))$.
Then as above one has that $w+E_{G_{n}}\subseteq E_{G}$ and
$w+I_{G_{n}}\subseteq I_{G}$.

We then have by Lemma 3.21 that $\overline{p}\in\bigcap_{G\in
D}\overline{I_{G}}$. Now $\Pi_{f\in F}\overline{B}$ is a neighborhood of
$\overline{p}$ so pick $w\in I_{F}\cap\Pi_{f\in F}\overline{B}$. Pick
$n,C_{1},C_{2},\\{G_{i}\\}_{i=1}^{n}$ and $\eta\in\Pi_{i\in C_{1}}G_{i}$ as
guaranteed by the fact that $w\in I_{F}$. Let $r=|C_{2}|$ and let
$h_{1},h_{2},\cdots,h_{r}$ be the elements of $C_{2}$ listed in increasing
order. Let $v=\sum_{i=1}^{r}m_{\frac{1}{k}}(G_{h_{i}})$. If $h_{1}=1$, let
$c(1)=\alpha(G_{1})(1)$. If $h_{1}>1$, let

$c(1)=\sum_{i=1}^{h_{1}-1}x(m_{\frac{1}{k}}(G_{i}),\alpha(G_{i}),\tau(G_{i}),\eta(i)+\alpha(G_{h_{1}})(1).$

For $1<j\leq m_{\delta}(G_{h_{1}})$ let $c(j)=\alpha(G_{h_{1}})(j)$ and for
$1\leq j\leq m_{\frac{1}{k}}(G_{h_{1}})$ let $\mu(j)=\tau(G{h_{1}})(j)$.

Now let $s\in\\{1,2,...,r-1\\}$ and let
$u=\sum_{i=1}^{s}m_{\frac{1}{k}}(G_{h_{i}})$ if $h_{s+1}=h_{s}+1$ let
$c(u+1)=\alpha(G_{h_{s}})(m_{\frac{1}{k}}(G_{h_{s}})+\alpha(G_{h_{s+1}})(1)$.
If $h_{s+1}>h_{s}+1$, let

$c(u+1)=\alpha(G_{h_{s}})(m_{\frac{1}{k}}(G_{h_{s}}+1)\\\
+(\sum_{i=h_{s}+1}^{h_{s+1}-1})x(m_{\frac{1}{k}}(G_{i}),\alpha(G_{i}),\tau(G_{i}),\eta(i))+\alpha(G_{h_{s+1}}(1)).\\\
$

And for $u<j\leq\sum_{i=1}^{s+1}m_{\frac{1}{k}}(G_{h_{i}})$, let
$\mu(j)=\tau(G_{h_{s+1}})(j-u)$.

If $h_{r}=n$, let $c(v+1)=\alpha(G_{n})(m_{\frac{1}{k}}({G_{n}+1}))$ if
$h_{r}<n,$ let

$c(v+1)=\alpha(G_{n})(m_{\frac{1}{k}}({G_{n}+1}))+\sum_{i=h_{r}}^{n}(x(m_{\frac{1}{k}}(G_{i}),\alpha(G_{i}),\tau(G_{i}),\eta(i))).$

Then $c\in S_{\frac{1}{k}}^{v+1}$, $M\in\mathcal{J}_{v}$ and for each $f\in
F$, $x(v,c,\mu,f)\in B$ as required. ∎

## References

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# Maximal Acceleration, Reciprocity & Nonlocality

Luca Buoninfante<EMAIL_ADDRESS>Department of Physics, Tokyo
Institute of Technology, Tokyo 152-8551, Japan

###### Abstract

The existence of a fundamental scale is expected to be a key feature of
quantum gravity. Many approaches take this property as a starting assumption.
Here, instead, we take a less conventional viewpoint based on a critical
inspection of both fundamental principles and kinematic laws. We point out
that rigorous arguments suggest a more urgent need to revise known theories to
incorporate a fundamental acceleration scale already in flat space. The
reciprocity principle can naturally do so. In addition to noticing links with
string theory, we argue that the reciprocity principle implies an infinite-
derivative generalization of the Einstein-Hilbert action that makes the
gravitational interaction fundamentally nonlocal, thus providing a guiding
principle that could lead us towards the formulation of a consistent theory of
quantum gravity.

Essay received an Honorable Mention in the 2021 Gravity Research Foundation
Essays Competition.

The three fundamental constants associated with the notions of relativity,
quantum and gravity are the speed of light $c,$ (reduced) Planck’s constant
$\hbar$ and Newton’s constant $G,$ respectively. By combining them together we
can define the Planck length $\ell_{\rm p}=\sqrt{\hbar G/c^{3}},$ or the
Planck mass $m_{\rm p}=\hbar/c\ell_{\rm p}.$ In general, these quantities are
not fundamental scales and are not related to any notion of discretization.
However, they acquire an important meaning when both quantum and gravitational
effects are non-negligible. For instance, $\ell_{\rm p}$ is interpreted as the
minimal observable length in any experiment because by probing smaller
distances one would form a black hole Garay:1994en . Thus, $m_{\rm p}$ is
expected to be the maximum mass for an elementary particle.

In a quantum-gravitational regime we can also define the following upper bound
on the gravitational acceleration, known as Planck acceleration:

$a_{\rm p}=\frac{Gm_{\rm p}}{\ell_{\rm p}^{2}}=\sqrt{\frac{c^{7}}{\hbar
G}}=\frac{m_{\rm p}c^{3}}{\hbar}\simeq 5.6\times 10^{52}\,{\rm m/s^{2}}\,,$
(1)

which corresponds to the gravitational acceleration on the surface (event
horizon) of the smallest and most concentrated object in Nature, i.e. a
Planck-size black hole.

These limitations seem to suggest that several conceptual problems – like
curvature singularities and ultraviolet divergences – could be solved.
However, in quantum general relativity $\ell_{\rm p}$ and $m_{\rm p}$ are not
physical cut-offs. Moreover, the above arguments only apply outside the black
holes; in fact, when approaching very short distances inside a black hole, the
accelerations may become larger. At those scales, to even address any question
one would require a consistent and predictive (possibly four-dimensional)
theory of quantum gravity that can naturally incorporate the above concepts,
e.g. the notion of minimal length or maximal acceleration.

Many quantum-gravity approaches take the existence of a fundamental scale as a
starting assumption, and only concern modifications and/or extensions of the
low-energy dynamics without paying much attention to kinematics. In this Essay
we wish to take a different viewpoint which is usually underestimated or less
appreciated, and that is based on a critical inspection of fundamental
principles, and on a possible generalization of the standard relativistic laws
of kinematics, which then will also be followed by a revision of the dynamical
laws (i.e. Lagrangians and field equations). This logical way of thinking
should be seen as a first necessary step towards the formulation of a
consistent theory of quantum gravity.

## Maximal Acceleration

Although we can agree that a physical cut-off scale should exist in Nature, it
is not clear what dimensionful scale is more fundamental; could it be a
length, a time or an acceleration,…?

Too many guesses can often lead on the wrong path. It is wiser to rely on what
we already know and do a deeper investigation of standard theories by looking
at things from a different angle. This viewpoint was taken by Caianiello
Caianiello:1981jq ; Caianiello-HUP  who was the first to understand that
simply by combining notions of special relativity and quantum mechanics
(without gravity) one can derive an upper limit to the proper acceleration of
any massive particle.

The starting point in Caianiello’s poof is the following Heisenberg
uncertainty relation Caianiello-HUP ; Wood:1989qu :

$\Delta E\,\Delta v(t)\geq\frac{\hbar}{2}\left|\frac{{\rm d}v(t)}{{\rm
d}t}\right|\,,$ (2)

where $\Delta E$ and $\Delta v(t)$ are the uncertainties on energy $E$ and
velocity $v(t)$ of a quantum particle of rest mass $m.$ By using the fact that
$\Delta v=(\langle v^{2}\rangle-\langle v\rangle^{2})^{1/2}\leq c,$ and
$\Delta E\leq E\leq mc^{2}$ (in the instantaneous particle rest frame), we
obtain the remarkable bound Caianiello-HUP

$a\leq\frac{2mc^{3}}{\hbar}\equiv a_{\rm max}\,,$ (3)

where $a={\rm d}v/{\rm dt}$ is the proper acceleration of the particle and
$a_{\rm max}$ is known as Caianiello’s maximal acceleration. Obviously, such a
bound is intrinsically quantum and relativistic, indeed in either limits
$c\rightarrow\infty$ or $\hbar\rightarrow 0$ we recover the classical scenario
with no limitation, i.e. $a\leq\infty.$

The bound (3) does not rely on any gravitational feature. However, as
mentioned above, in a quantum-gravitational regime we would expect $m\leq
m_{\rm p}$ for any elementary particle. Therefore, when gravitational effects
are taken into account one has

$m\leq m_{\rm p}\quad\Rightarrow\quad a\leq\frac{2m_{\rm
p}c^{3}}{\hbar}=2a_{\rm p}\,,$ (4)

which would represent a fundamental upper bound on the acceleration of any
elementary massive particle, and coincides (up to a factor $2$) with the
Planck acceleration in Eq. (1). Caianiello’s relation (3) also implies an
upper bound on the force that any elementary particle of mass $m$ can feel,
$F\leq 2m^{2}c^{3}/\hbar\leq 2m_{\rm p}^{2}c^{3}/\hbar=2c^{4}/G.$ The last
part of the inequality coincides up to a factor of order one with the so
called principle of maximum tension postulated by Gibbons Gibbons:2002iv .

Hence, in the quantum-gravitational regime a physical cut-off with dimensions
of acceleration (or force) is expected to emerge. This may suggest the more
urgent need to find a theory that naturally incorporates such a new scale,
even before bothering about a quantum theory of gravity.

## Reciprocity Principle

Any new progress in physics should always be supported by some guiding
principle. We should indeed ask – is there any fundamental principle that can
be advocated in order to naturally incorporate a new fundamental scale in our
theory? The answer is YES!

In 1938, Born realized that physical laws may be invariant under the so called
reciprocity111The word “reciprocity” was chosen in analogy with the
“reciprocal lattice ” in the theory of crystals, where the motion of the
particle can be mapped from the $x$-coordinate space to the $p$-momentum space
Born0 . transformation Born0

$x^{\mu}\rightarrow\alpha\,p^{\mu}\quad{\rm and}\quad
p^{\mu}\rightarrow-\frac{1}{\alpha}\,x^{\mu}\,,$ (5)

where the positive constant $\alpha$ has dimensions $[{\rm kg}^{-1}\,{\rm s}]$
or $[{\rm m}\,{\rm s}^{-1}\,{\rm N}^{-1}].$ Some physical equations that are
manifestly invariant under (5) are the Hamilton equations, the commutation
relations, the angular momentum components etc. Born promoted the invariance
of physical laws under (5) to a symmetry principle known as reciprocity
principle. Initially, he aimed at constructing a new theory that could predict
the meson mass spectrum Born , but his results were invalidated by
experiments. However, the validity of the reciprocity principle can not be
discarded yet Morgan .

The first main observation made by Born was that the four-dimensional line
element ${\rm d}s^{2}=\eta_{\mu\nu}{\rm d}x^{\mu}{\rm d}x^{\nu},$ with
$\eta_{\mu\nu}={\rm diag}(-1,1,1,1),$ is not invariant under (5), and that
this was an important issue to solve even before trying to combine quantum
mechanics and general relativity. Thus, he proposed the following generalized
eight-dimensional line element which is invariant under (5):

${\rm d}s^{2}=\eta_{\mu\nu}{\rm d}x^{\mu}{\rm
d}x^{\nu}+\alpha^{2}\eta_{\mu\nu}{\rm d}p^{\mu}{\rm
d}p^{\nu}\,,\quad\alpha=\frac{c^{4}}{\mathcal{A}^{2}\hbar}\,,\,\,$ (6)

where $\mathcal{A}$ is a fundamental acceleration scale and it represents a
maximal value on the acceleration of any massive particles moving along their
worldline; this can be easily shown by using $|{\rm d}\vec{x}/{\rm d}t|\leq c$
and the timelike condition ${\rm d}s^{2}<0$ Caianiello:1981jq . Note that, we
could have written $\alpha$ in terms of a new fundamental length, mass or
force scale too, but we have chosen to treat acceleration (i.e. kinematics) as
more fundamental in view of our arguments above.

We would expect to recover the four-dimensional line element not only in the
limit of no upper bound ($\mathcal{A}\rightarrow\infty$) but also when
$c\rightarrow\infty$ or $\hbar\rightarrow 0.$ This implies that $\mathcal{A}$
itself must have a certain dependence on $c$ and $\hbar$ in order to recover
the correct “classical” limit, thus suggesting a proportionality relation
between $\mathcal{A}$ and the Planck acceleration $a_{\rm p}$ introduced in
Eq. (1).

Hence, we have shown that the reciprocity principle implies the existence of a
fundamental acceleration scale which corresponds to a maximal limiting value
on any acceleration in Nature, consistently with the aforementioned concepts
pioneered by Caianiello in Caianiello:1981jq ; Caianiello-HUP . It is worth
mentioning that Caianiello considered a similar line element in his
geometrical formulation of quantum mechanics quantum-geometry ;
Torrome:2018zck  but he chose the coordinates
$\big{(}x^{\mu},\frac{c^{2}}{\mathcal{A}}\frac{{\rm d}x^{\mu}}{{\rm
d}s}\big{)}.$

## Nonlocality

The reciprocity principle has far-reaching unexpected consequences, among
which the most interesting is that point-like interactions become
intrinsically nonlocal. Born applied the principle to quantum electrodynamics
and showed that the field equations (in Lorenz gauge) for the potential
$A_{\mu}$ sourced by $J_{\mu}$ are modified as follows Born ; Born-nonlocal

$e^{-\ell^{2}\Box}\Box A_{\mu}=4\pi
J_{\mu}\,,\quad\ell=\frac{c^{2}}{\mathcal{A}}\,,$ (7)

where $\Box=\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}$ is the flat
d’Alembertian and $\ell$ is a length scale at which nonlocal effects become
relevant. The infinite-order derivatives in $e^{-\ell^{2}\Box}$ can regularize
the singularity of Coulomb potential at $r=0$ Born .

The emergence of nonlocality222Up to our knowledge, this type of nonlocality
and non-polynomial differential operator were introduced independently of
reciprocity for the first time in 1931 by Born himself Born-rumer . is
unavoidable if the reciprocity principle is valid, as clearly stated by Born
in Born : “…in the reciprocity theory it is a necessity.” Nonlocal field
theories were further investigated by Pais Pais:1950za , Efimov Efimov:1967pjn
, and in a slightly different version by Yukawa Yukawa:1950eq  .

We now wish to go beyond Born’s pioneering work, and argue that the same
nonlocal modification happens if we apply the reciprocity principle to gravity
progress . The gravitational analogue of Eq. (7) is given by the linearized
field equations for the metric perturbation around Minkowski,
$h_{\mu\nu}=g_{\mu\nu}-\eta_{\mu\nu},$ written in de Donder gauge:

$e^{-\ell^{2}\Box}\Box\left(h_{\mu\nu}-\frac{1}{2}\eta_{\mu\nu}h\right)=-16\pi
G\,T_{\mu\nu}\,,$ (8)

where $h=\eta^{\mu\nu}h_{\mu\nu}$ and $T_{\mu\nu}$ is the stress-energy tensor
of matter. Remarkably, exactly the same type of “nonlocal gravity" has been
investigated intensively in the last decades in relation to the problems of
renormalizability, unitarity and singularities in quantum gravity Krasnikov ;
Biswas:2005qr .

Analogously to the case of Coulomb’s potential, nonlocality modifies the
short-distance behavior of Newton’s potential which, in presence of a point-
like source
$T_{\mu\nu}=M\delta_{\mu}^{0}\delta_{\nu}^{0}\delta^{(3)}(\vec{r}),$ reads

$\Phi(r)=-\frac{1}{2}h_{00}(r)=-\frac{GM}{r}{\rm
Erf}\left(\frac{r}{2\ell}\right)\,,$ (9)

and it is indeed regular at the origin, $\Phi(r=0)=-GM/(\ell\sqrt{\pi}).$

Furthermore, one can show that the minimal form of the gravitational action
compatible with the field equations (8), and therefore with the reciprocity
principle, must be quadratic in the curvature invariants, and it is given by

$S=\frac{1}{16\pi G}\int{\rm
d}^{4}x\sqrt{-g}\left(R+G_{\mu\nu}\frac{e^{-\ell^{2}\Box}-1}{\Box}R^{\mu\nu}\right)\,,$
(10)

where $R_{\mu\nu}$ is the Ricci tensor, $R$ the Ricci scalar and $G_{\mu\nu}$
the Einstein tensor.

Figure 1: Schematic illustration summarizing the intriguing relations among
reciprocity principle, maximal acceleration and nonlocality.

## Remarks & Outlook

In Fig. 1 we have schematically summarized the content of this work. Let us
now make some final important remarks.

  * •

The scale $\mathcal{A}$ represents a fundamental limit on the acceleration. If
the weak equivalence principle is valid, then an upper bound on the
gravitational acceleration (or force) must exist. This feature may be related
to the fact that the gravitational potential (8) is bounded. Similar
singularity avoidances due to a maximal acceleration were also found in
Caianiello’s theory Feoli:1999cn  and Loop Quantum Gravity Rovelli:2013osa .

  * •

The fundamental scale must be an invariant, i.e. the same for any observer,
therefore a revision of the transformation laws in relativistic kinematics may
be required. A generalized transformations’ group that leaves invariant both
an eight-dimensional line element and a maximal acceleration was found in
Scarpetta:1984nn .

  * •

A maximal acceleration implies that any would-be point-like object is instead
extended Caianiello:1981is . The same interplay characterizes string theory,
where the extended nature of strings implies a maximal acceleration
Frolov:1990ct . Other features shared with string theory are the nonlocal
operator $e^{-\ell^{2}\Box}$ Sen:2004nf ; Tseytlin:1995uq , and a connection
between reciprocity and T-duality Veneziano:1986zf .

  * •

Nonlocal theories of gravity Krasnikov ; Biswas:2005qr  are often criticized
for being too arbitrary and not motivated from first principles. On the
contrary, we argued, for the first time, that the reciprocity principle
implies that gravity is fundamentally nonlocal, and uniquely selects one
single form of nonlocality given by the exponential in Eqs. (8) and (10).

In this Essay we pointed out that quantum-relativistic arguments suggest a
critical revision of first principles and standard kinematic laws to
incorporate a new fundamental acceleration scale. The reciprocity principle
can naturally do so, and as a consequence a nonlocal generalization of
dynamical laws emerges. Future investigations are necessary to inspect the
validity of the reciprocity principle and the existence of the maximal bound
on acceleration, especially in relation to the formulation of a consistent
quantum theory of gravity progress .

### Acknowledgements

I dedicate this work to the memory of Eduardo R. Caianiello whom I was not
lucky enough to meet. I acknowledge financial support from JSPS and KAKENHI
Grant-in-Aid for Scientific Research No. JP19F19324.

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