2 Basic results

In this section, we present some preliminary concepts, definitions, theorems, conventions and results that will be used frequently later in this paper.

First, we give a brief introduction of the Stone–Čech compactification of a discrete semigroup. For any discrete semigroup $(S,\cdot )$ , the Stone–Čech compactification $\beta S$ of the discrete semigroup $(S,\cdot )$ is the set of all ultrafilters on $S$ . The principal ultrafilters are identified with the points of $S$ . For a subset $A$ of $S$ , we denote by $\bar{A}$ the set ${\bar {A}=\{ p\in \beta S : A\in p\}}$ . Then, the set $\{ \bar {A} : A\subseteq S\}$ forms a clopen basis for a topology on $\beta S$ . One can extend the operation $\cdot $ on $S$ to the Stone–Čech compactification $\beta S$ of $S$ , which makes $(\beta S,\cdot )$ a compact, right topological semigroup (meaning that for any $p\in \beta S$ , the function $\rho _p:~\beta S \rightarrow \beta S$ defined by $\rho _p(q)=q\cdot p$ is continuous) with $S$ contained in its topological centre (meaning that for any $x\in S$ , the function $\lambda _x : \beta S \rightarrow \beta S$ defined by $\lambda _x (q)=x\cdot q$ is continuous). For $p,q \in \beta S $ and $A \subseteq S$ , we have $A \in p\cdot q$ if and only if $\{ x \in S :x^{-1}A \in q\} \in p$ , where $x^{-1}A=\{ y \in S: x\cdot y \in A \}$ .

A nonempty subset $I$ of a semigroup $(T,\cdot )$ is a left ideal of $S$ if $T\cdot I \subseteq I$ , a right ideal of $S$ if $I\cdot T \subseteq I$ , and a two-sided ideal (or simply an ideal) if it is both a left and a right ideal. A left ideal is minimal if it does not contain any proper left ideal. In a similar way, one can define a minimal right ideal and the smallest ideal. For any compact Hausdorff right topological semigroup $(T,\cdot )$ , there is a smallest two-sided ideal $K(T)$ :

$$ \begin{align*} K(T) & = \bigcup\{L:L \text{ is a minimal left ideal of } T\} \\ & = \bigcup\{R:R \text{ is a minimal right ideal of } T\}. \end{align*} $$

For a given minimal left ideal $L$ and minimal right ideal $R$ , $L\cap R$ is a group and, hence, it contains an idempotent (an element $a\in S$ is an idempotent if $a = a\cdot a$ ). An idempotent belonging to the smallest ideal is minimal and conversely. We use $\mathcal {P}_{f}(X)$ to denote the set of all finite nonempty subsets of a set $X$ and $E(S)$ to denote the set of all idempotents of $S$ . For more details, see [Reference Hindman and Strauss16, Section 2.2].

Our results are derived for an arbitrary subsemigroup of $\mathbb R$ that is dense in $(0, \infty )$ , for example, the set of positive rational numbers. In [Reference Hindman and Leader15], Hindman and Leader introduced the semigroup consisting of ultrafilters converging to zero. For a dense subsemigroup $S$ of $((0, \infty ), +)$ , define $ 0^+(S)= \{p\in \beta S: (0, \epsilon )\cap S\in p \text { for all } \epsilon>0\}$ . It is necessary to take $S$ as a dense subsemigroup of $((0, \infty ), +)$ ; for otherwise, $ 0^+(S)$ will be empty.

As $0^+(S)$ is a compact Hausdorff right topological semigroup, by [Reference Berglund, Junghenn and Milnes5, Theorem 1.3.11], $0^+(S)$ contains a smallest two-sided ideal, denoted by $K(0^+(S))$ , and it must contain an idempotent by [Reference Ellis8, Corollary 2.10]. As $ 0^+(S)$ is a two-sided ideal of $(\beta (0,1)_d, \cdot )$ , we have $K(\beta (0,1)_d, \cdot )\subseteq 0^+(S)$ and, therefore, by [Reference Hindman and Strauss16, Theorem 1.65], $K(0^+(S), \cdot )=K(\beta (0,1)_d, \cdot )$ . Further, $0^+(S)\cap K(\beta S)=\emptyset $ and, hence, $0^+(S)$ provides some new information that is not available from $K(\beta S)$ . After introducing the notion of a semigroup of ultrafilters near zero or converging to $0$ , Hindman and Leader studied some Ramsey theoretic results near zero in [Reference Hindman and Leader15]. In [Reference Patra and Shaikh20], Patra and Shaikh gave the name HL semigroups to the semigroups satisfying the hypotheses of Theorem 2.1.

In an HL semigroup, the requirement $y < 1$ is not essential, but makes the proofs simpler since under that assumption, if $x < 1/n$ , then $yx < 1/n$ .

In [Reference Furstenberg9], Furstenberg introduced a number of classes of large sets originating from topological dynamics. Alternative characterisations of such sets are also available in terms of the algebraic structure of $\beta \mathbb {N}$ . Next, we introduce the IP sets and central sets which have rich combinatorial properties. There is an intimate relation between IP sets and idempotents. We need the following definitions to move forward.

The next definition gives the near-zero version of the IP-set in $\mathbb N$ . The IP-set near zero is an extremely important object that will be used to prove our results in the next section.

The following theorem shows that IP sets near zero are members of idempotents near zero.

The next definition gives the near-zero version of a central set.

In [Reference Bergelson and Glasscock4], Bergelson and Glasscock investigated the interplay between additive and multiplicative largeness. The following theorem shows that combined additive and multiplicative structure can be found in one partition cell for any finite partition of $(S\cap (0,1), \cdot )$ .

As a consequence of the above theorem, we have the following corollary.

3 Product and translated product near zero

In [Reference Goswami10], Goswami proved that $\{a, b, ab, b(a+1)\}$ is monochromatic for every finite colouring of $\mathbb N$ . His proof is similar to the argument of Moreira [Reference Moreira18], who employed van der Waerden’s theorem [Reference van der Waerden23], while Goswami made iterative use of Hindman’s theorem. In this section, our main target is to show that $\{a, b, ab, b(a+1)\}$ is monochromatic near zero for dense subsemigroups of $((0, \infty ), +)$ , which means we have to show that for any finite partition of $S\cap (0,1)$ , at least one partition cell contains $a, b, ab, b(a+1)$ . We shall follow similar arguments to those used by Goswami in [Reference Goswami10]. Before going to our main theorem, we need the following definition.

Definition 3.1. Let $S$ be an HL semigroup and let $A\subseteq S$ . Then, we denote

$$ \begin{align*}-y+A=\{x\in S: x+y\in A\} \quad\text{and}\quad y^{-1}A=\{x\in S: xy\in A\}.\end{align*} $$

We need the following three lemmas to prove our main theorem. The first is a near-zero version of a reformulated Hindman’s theorem [Reference Hindman12].

Next, we state Goswami’s theorem and its near-zero version.

4 Sums and quotients near zero

In the previous section, we proved our results using combinatorial arguments. In this section, we give stronger results by an algebraic approach using tensor products of ultrafilters. We begin by recalling the definition of a tensor product.

Definition 4.1 [Reference Hindman and Strauss17, Definition 1.1].

Let $X$ and $Y$ be discrete spaces. Let $p\in \beta X$ and $q\in \beta Y$ . Then, the tensor product of $p$ and $q$ is defined by

$$ \begin{align*}p\otimes q=\{A\subseteq X\times Y: \{x\in X: \{y\in Y: (x, y)\in A\}\in q\}\in p\}.\end{align*} $$

It is easy to see that $ A\times B\in p\otimes q$ if and only if $A\in p$ and $B\in q$ . Next, we define a mapping that connects the tensor product with $0^+(S)$ for an HL semigroup.

Definition 4.2. Let $S$ be an HL semigroup and $s_0\in S$ be a fixed element of $S$ . Define a map

$$ \begin{align*}D: S\times S\to S \quad\text{by}\quad D(x, y)=\begin{cases} {y}/{x} & \text{if } x\in S\cap (0, 1), \\ s_0 &\text{otherwise}. \end{cases}\end{align*} $$

The following lemma shows that the image of a tensor product of $u\in 0^+(S)$ with itself under $D$ is in $0^+(S)$ .

Proof. Let $\epsilon>0$ be given. Now,

$$ \begin{align*} D^{-1} & [S\cap (0, \epsilon)] =\{(x, y)\in S\times S: D(x, y)\in S\cap (0, \epsilon)\} \\ & \supseteq\{(x, y)\in S\times S: {y}/{x}\in S\cap (0, \epsilon)\}=\{(x, y)\in S\times S: y\in x(S\cap (0, \epsilon))\}. \end{align*} $$

From the definition, $\{(x, y)\in S\times S: y\in x(S\cap (0, \epsilon ))\}\in u\otimes u$ if and only if

$$ \begin{align*}\{x\in S: \{y\in S: y\in x(S\cap (0, \epsilon))\}\in u\}\in u,\end{align*} $$

and this holds if and only if $\{x\in S: \{ x(S\cap (0, \epsilon ))\}\in u\}\in u$ . Since $x(S\cap (0, \epsilon ))\in u$ for each $x\in S\cap (0, 1)$ , we have $D^{-1}[S\cap (0, \epsilon )]\in u\otimes u$ . Thus, $S\cap (0, \epsilon )\in D(u\otimes u)$ for all $\epsilon>0$ and so $D(u\otimes u)\in 0^+(S)$ .

We write $F<G$ for any $F, G\in P_f(\mathbb N)$ if $\max F < \min G$ .

We are now in a position to prove the main result of this section. The proof is adapted from the proof of [Reference Di Nasso, Luperi Baglini, Mennuni, Ragosta and Vegnuti19, Theorem 1.4].

Proof. Since $C\in D(u\otimes u)$ , we have $D^{-1}[C]\in u\otimes u$ . By Lemma 4.4, there is a convergent sequence $\langle {z_n}\rangle _{n=1}^{\infty }$ with $\langle {{z_{n+1}}/{z_n}}\rangle _{n=1}^{\infty }$ convergent such that when ever $F, G\in P_f(\mathbb N) \text{and} F<G$ , we have $(\sum _{n\in F}z_n, \sum _{n\in G}z_n)\in D^{-1}[C]$ and so $D(\sum _{n\in F}z_n, \sum _{n\in G}z_n)={\sum _{n\in G}z_n}/{\sum _{n\in F}z_n}\in C$ .

Now, set $x_n={z_{n+1}}/{z_1}$ and $y_n={z_{n+1}}/{z_n}$ . Both the sequences $\langle {x_n}\rangle _{n=1}^{\infty }$ and $\langle {y_n}\rangle _{n=1}^{\infty }$ are convergent. Also, for every $F, G\in P_f(\mathbb N)$ , we have $\sum _{n\in F}x_n={\sum _{n\in F}z_{n+1}}/{z_1}$ and

$$ \begin{align*}\dfrac{\sum_{n\in G}x_{n}}{\sum_{n\in F}x_{n}}=\dfrac{\sum_{n\in G}z_{n+1}/z_1}{\sum_{n\in F}z_{n+1}/z_1}=\dfrac{\sum_{n\in G}z_{n+1}}{\sum_{n\in F}z_{n+1}}.\end{align*} $$

It follows that $\text {FS} (\langle {x_n}\rangle _{n=1}^{\infty })\cup \{{\sum _{n\in G}x_n}/{\sum _{n\in F}x_n}: F, G\in P_f(\mathbb N) \text { and } F<G\}\subseteq C$ . Moreover, if $F\in P_f(\mathbb N)$ and $k\leq \min F$ , analogously,

$$ \begin{align*} \sum_{l\in F}\prod_{n=k}^ly_n=\sum_{l\in F}\prod_{n=k}^l\frac{z_{n+1}}{z_n}=\sum_{l\in F}\frac{z_{l+1}}{z_k}=\frac{\sum_{l\in F}z_{l+1}}{z_k}\in C.\\[-40pt] \end{align*} $$

The following corollary shows that the pattern $x, y, x+y, {y}/{x}$ is partition regular near zero.

Note that Goswami’s theorem near zero follows from Corollary 4.7, by taking $a=x, b={y}/{x}$ .

In 2011, Hindman proved that for every finite colouring of $\mathbb N$ , there must exist $a, b, c$ and $d$ in $\mathbb N$ such that $a, b, c$ and $d$ are distinct and the colour of $a+b=cd$ is the same as that of $a, b, c$ and $d$ [Reference Hindman14]. In 2020, Patra and Shaikh proved this result near zero [Reference Patra and Shaikh20]. In 2017, J. Moreira proved that for every finite colouring of $\mathbb N$ , there exist $a, b\in \mathbb N$ such that the set $\{a, a\cdot b, a+b\}$ is monochromatic [Reference Moreira18]. We end this work with the following question.