ERDŐS–LIOUVILLE SETS

Abstract

In 1844, Joseph Liouville proved the existence of transcendental numbers. He introduced the set $\mathcal L$ of numbers, now known as Liouville numbers, and showed that they are all transcendental. It is known that $\mathcal L$ has cardinality $\mathfrak {c}$, the cardinality of the continuum, and is a dense $G_{\delta }$ subset of the set $\mathbb {R}$ of all real numbers. In 1962, Erdős proved that every real number is the sum of two Liouville numbers. In this paper, a set W of complex numbers is said to have the Erdős property if every real number is the sum of two numbers in W. The set W is said to be an Erdős–Liouville set if it is a dense subset of $\mathcal {L}$ and has the Erdős property. Each subset of $\mathbb {R}$ is assigned its subspace topology, where $\mathbb {R}$ has the euclidean topology. It is proved here that: (i) there exist $2^{\mathfrak {c}}$ Erdős–Liouville sets no two of which are homeomorphic; (ii) there exist $\mathfrak {c}$ Erdős–Liouville sets each of which is homeomorphic to $\mathcal {L}$ with its subspace topology and homeomorphic to the space of all irrational numbers; (iii) each Erdős–Liouville set L homeomorphic to $\mathcal {L}$ contains another Erdős–Liouville set $L'$ homeomorphic to $\mathcal {L}$. Therefore, there is no minimal Erdős–Liouville set homeomorphic to $\mathcal {L}$.

1 Introduction

It has been known for over 175 years that every Liouville number is transcendental and for 120 years that the set $\mathcal {L}$ of Liouville numbers is uncountable. Notwithstanding this, the set $\mathcal {L}$ is known to have Lebesgue measure zero. So in this sense, $\mathcal {L}$ is very small. Therefore, it is surprising that each real number equals the sum of two Liouville numbers. It is reasonable to ask if $\mathcal {L}$ is the smallest set, in some sense, with this property. In this paper, it is proved that there is an uncountable number of sets smaller than $\mathcal {L}$ which have this property. Indeed, there are $2^{\mathfrak {c}}$ such subsets of $\mathcal {L}$ no two of which are homeomorphic as subspaces of $\mathbb {R}$ .

2 Preliminaries

Remark 2.1. In 1844, Joseph Liouville proved the existence of transcendental numbers [2, 3]. He introduced the set $\mathcal L$ of real numbers, now known as Liouville numbers, and showed that they are all transcendental. A real number x is said to be a Liouville number if for every positive integer n, there exists a pair of integers $(p,q)$ with $q>1$ such that

$$ \begin{align*} 0< \bigg\lvert x-\frac{p}{q}\bigg\rvert<\frac{1}{q^n}. \end{align*} $$

This definition of a Liouville number can be reformulated as follows. For a given irrational x, let $p_k/q_k=p_k(x)/q_k(x)$ , where $q_k(x)>0$ , denote the sequence of convergents of the continued fraction expansion of x; then for every $n\in \mathbb {N}$ , there are infinitely many k such that $q_{k+1}>q_k^n$ . A more restrictive class of Liouville numbers is obtained by requiring this inequality to hold for every $k> N=N(n)\in \mathbb {N}$ . Such numbers are called strong Liouville numbers.

In 1962, Erdős [8] proved that every real number is the sum of two Liouville numbers (and also the product of two Liouville numbers). He gave two proofs. One was a constructive proof. The other proof used the fact that the set $\mathcal {L}$ of all Liouville numbers is a dense $G_{\delta }$ -set in $\mathbb {R}$ and showed that every dense $G_{\delta }$ -set in $\mathbb {R}$ has this property.

Definition 2.2. A set W of complex numbers is said to have the Erdős property if every real number is a sum of two numbers in W.

Remark 2.3. Recall that if A and B are subsets of the set $\mathbb {C}$ of all complex numbers, then the sum-set is defined to be $A+B=\{a+b:a\in A, b\in B\}$ . So the subset W of $\mathbb {C}$ has the Erdős property if the sum-set $W+W$ contains the set $\mathbb {R}$ . (See [4, 13].)

Remark 2.4. By the theorem proved by Erdős mentioned above, the set $\mathcal {L}$ of all Liouville numbers has the Erdős property.

Remark 2.5. If W is a set with the Erdős property, then every set containing W also has the Erdős property.

Definition 2.6. A set W is said to be an Erdős–Liouville set if it has the Erdős property and is a dense subset of the set $\mathcal {L}$ of Liouville numbers.

Remark 2.7. It is not immediately obvious that there exist any Erdős–Liouville sets other than the set $\mathcal {L}$ itself. It is known that some sets of positive Lebesgue measure have the Erdős property, but they are not subsets of $\mathcal {L}$ as the set $\mathcal {L}$ is known to have measure zero. (See, for example, [5].) According to Petruska, [12], Erdős asked if the set of strong Liouville numbers has the Erdős property. However, Petruska [12] proved that it does not. He did this by showing that the sum of two strong Liouville numbers is either a Liouville number or a rational number. Hence, the sum of two strong Liouville numbers cannot equal any irrational number other than a Liouville number. However, it is proved in [7], in the text following Corollary 1.4 and in Section 3, that there does exist another Erdős–Liouville set. In [10], the set of ultra-Liouville numbers is introduced and it is shown that this set is a dense $G_{\delta }$ -subset of $\mathcal {L}$ which is therefore an Erdős–Liouville set.

Remark 2.8. In the literature, there are various strengthenings of the Erdős result on Liouville numbers. We mention explicitly [1, 14, 15]. The paper [9] shows that the set of Liouville numbers has a property stronger than the Erdős property. Though we do not study such properties, we record here that the $\mathfrak {c}$ Erdős–Liouville sets we produce in Theorem 4.6 also possess this stronger property, while Theorem 3.6 and the proof of Theorem 4.6 show that there are only $\mathfrak {c}$ dense $G_{\delta }$ subsets of $\mathbb {R}$ . The relevant theorem from [9] describing this stronger property is the following result.

Theorem 2.9. Let G be a dense $G_{\delta }$ -subset of $\mathbb {R}$ , I an interval in $\mathbb {R}$ with nonempty interior, and f a continuous function from I to $\mathbb {R}$ which is nowhere locally constant. (This means that f is not constant on any nonempty open subinterval of I.) Then there exists an $x\in G\cap I$ such that $f(x)\in G$ . Indeed, there is an uncountable number of such x.

If we put $f(x)=r-x$ , for $r,x\in \mathbb {R}$ and $I=\mathbb {R}$ , we see that f satisfies the conditions of the theorem and thus G has the Erdős property. However, as observed in [9], if we put $I=(0,\sqrt {r})$ and $f(x) = \sqrt {r-x^2}$ , we see that for every Erdős–Liouville set G, every positive real number is the sum of two squares of numbers in G. Also, the argument in [9, pages 63–64] with $L^1= \{\exp (\alpha ): \alpha \in \mathcal {L}\}$ leads to the observation that $L^1\cap {\mathcal {L}}$ is an Erdős–Liouville set. Although it was not explicitly mentioned in [9], it follows by induction that if $L^n= L^{n-1}\cap \mathcal {L}$ , for $n\in \mathbb {N}$ , $n>1$ , then each $L^n$ is an Erdős–Liouville set. However, we do not know if the sets $L^n$ are distinct from each other and distinct from $\mathcal {L}$ .

Proposition 2.10. Let S be a set of real numbers such that $W_1\supset S\supset W_2$ , where $W_1$ and $W_2$ are Erdős–Liouville sets. Then S is an Erdős–Liouville set.

Proof. As $S\supset W_2$ , by Remark 2.5, it has the Erdős property. Also as $W_2$ is dense in $\mathbb {R}$ , so too is S. Finally, as $S\subset W_1$ , it is a subset of $\mathcal {L}$ . Therefore, S is an Erdős–Liouville set.

3 Some topology

Before proving the existence of an uncountable number of Erdős–Liouville sets, we need to record some topology, some of which was laid bare in [5, 6, 11].

Definition 3.1. A topological space X is said to be topologically complete (or completely metrisable) if the topology of X is the same as the topology induced by a complete metric on X.

Of course, every complete metric space is topologically complete.

We denote by $\mathbb {P}$ the set of all irrational real numbers with the topology it inherits as a subspace of the euclidean space $\mathbb {R}$ .

A beautiful characterisation of the topological space $\mathbb {P}$ is given in [16, Theorem 1.9.8].

Theorem 3.2. The space of all irrational real numbers $\mathbb {P}$ is topologically the unique nonempty, separable, metrisable, topologically complete, nowhere locally compact, and zero-dimensional space. $\Box $

This has a Corollary 3.3, [16, Corollary 1.9.9], which is often proved using continued fractions.

Corollary 3.3. The space $\mathbb {P}$ is homeomorphic to the Tychonoff product $\mathbb {N}^{\aleph _0}$ of a countably infinite number of homeomorphic copies of the discrete space $\mathbb {N}$ of positive integers. Hence, $\mathbb {P}\times \mathbb {P}$ is homeomorphic to $\mathbb {P}$ . Indeed, $\mathbb {P}$ is homeomorphic to $\mathbb {P}^{\aleph _0}$ .

Remark 3.4. Recall that a subset X of a topological space Y is said to be a $G_{\delta }$ -set if it is a countable intersection of open sets in Y while X is said to be an $F_{\sigma }$ -set if it is a countable union of closed sets in Y. Obviously, a subset X of a topological space Y is a $G_{\delta }$ -set if and only if its complement is an $F_{\sigma }$ -set. We see immediately that in a metric space such as $\mathbb {R}$ , the set $\mathcal {T}$ of all transcendental real numbers is a $G_{\delta }$ -set as its complement is the countably infinite set $\mathbb {A}$ of all real algebraic numbers.

Now we connect the notion of $G_{\delta }$ -set in $\mathbb {R}$ to the property of being topologically complete.

Theorem 3.5 [16, Theorem A.63].

A subset of a separable metric topologically complete space is a $G_{\delta }$ -set in that space if and only if it is topologically complete.

Using Theorems 3.2, 3.5 and Corollary 3.3, we obtain the following result.

Theorem 3.6. Every $G_{\delta }$ subset of the set $\mathbb {P}$ of all irrational real numbers is homeomorphic to $\mathbb {P}$ and to $\mathbb {N}^{\aleph _0}$ . In particular, the space $\mathcal {T}$ of all real transcendental numbers and the space $\mathcal {L}$ of all Liouville numbers, with their subspace topologies from $\mathbb {R}$ , are both homeomorphic to $\mathbb {P}$ and to $\mathbb {N}^{\aleph _0}$ .

These results and a similar one [16, Theorem 1.9.6] characterising the space $\mathbb {Q}$ of all rational numbers with its euclidean topology, are used in [6, 11] to describe transcendental groups and topological transcendental fields.

4 The existence of $2^{\mathfrak {c}}$ Erdős–Liouville sets

Theorem 4.1. Let X be a topological space homeomorphic to $\mathbb {P}$ . Then X has a dense $G_{\delta }$ -set Y which is homeomorphic to $\mathbb {P}$ such that the cardinality of the set $X\setminus Y$ is $\mathfrak {c}$ , the cardinality of the continuum.

Proof. Consider the topological space $\mathcal {T}$ of all real transcendental numbers and the topological space $\mathcal {L}$ of all Liouville numbers. We saw in Corollary 2.6 and Remark 2.1 that $\mathcal {L}$ is a dense $G_{\delta }$ -set, and $\mathcal {T}$ and $\mathcal {L}$ are homeomorphic to $\mathbb {P}$ . Further, the cardinality of the set $\mathcal {T}\setminus \mathcal {L}$ is $\mathfrak {c}$ . As the properties of being a dense $G_{\delta }$ -set and having cardinality $\mathfrak {c}$ are preserved by homeomorphisms, the theorem is proved.

By Theorem 4.1 and Remark 2.1, we have the following corollary.

Corollary 4.2. The space $\mathcal {L}$ of all Liouville numbers has a dense $G_{\delta }$ -set $L_1 $ homeomorphic to $\mathcal {L}$ . Further, $L_1$ is an Erdős–Liouville set.

Theorem 4.3. If L is any Erdős–Liouville set homeomorphic to $\mathbb {P}$ , then it has a proper subset $L_1$ which is an Erdős–Liouville set homeomorphic to $\mathbb {P}$ . Therefore, there is no minimal Erdős–Liouville set homeomorphic to $\mathbb {P}$ . $\Box $

Our next theorem follows immediately from Corollary 4.2 and Theorem 4.1.

Theorem 4.4. There exist Erdős–Liouville sets $L_1, L_2,\ldots , L_n, \ldots, $ for $n\in \mathbb {N}$ , such that

$$ \begin{align*} \mathcal{L}\supset L_1\supset L_2\supset\cdots\supset L_n\supset\cdots \end{align*} $$

with each $L_n\setminus L_{n+1}$ having cardinality $\mathfrak {c}$ and each $L_{n+1}$ a $G_{\delta }$ -set in $L_n$ which is homeomorphic to $\mathbb {P}$ . $\Box $

Theorem 4.5. There exist $2^{\mathfrak {c}}$ Erdős–Liouville sets no two of which are homeomorphic.

Proof. First, we note that there are precisely $2^{\mathfrak {c}}$ subsets of the set $\mathcal {L}$ of all Liouville numbers as $\mathcal {L}$ has cardinality $\mathfrak {c}$ . So the cardinality of the set of Erdős–Liouville sets is not greater than $2^{\mathfrak {c}}$ .

Using the notation of Theorem 4.4, let W be any subset of $\mathcal {L}\setminus L_1$ . As $L_1$ is an Erdős–Liouville set and $L_1\subset \mathcal {L}$ , Remark 2.5 implies that $L_1\cup W$ is an Erdős–Liouville set. As there are $2^{\mathfrak {c}}$ subsets W of the set $\mathcal {L}\setminus L_1$ , it follows that there are $2^{\mathfrak {c}}$ distinct Erdős–Liouville sets. So it remains to show only that amongst these, there are $2^{\mathfrak {c}}$ no two of which are homeomorphic.

By the Laverentieff theorem, [16, Theorem A8.5], there are at most $\mathfrak {c}$ subspaces of $\mathbb {R}$ which are homeomorphic. As there are $2^{\mathfrak {c}}$ distinct Erdős–Liouville sets, it follows that there are $2^{\mathfrak {c}}$ Erdős–Liouville sets no two of which are homeomorphic, as required.

Theorem 4.6. There exist $\mathfrak {c}$ Erdős–Liouville sets each of which is homeomorphic to $\mathcal {L}$ with its subspace topology. So each is homeomorphic to $\mathbb {P}$ .

Proof. Using the notation of Theorem 4.4, $\mathcal {L}\supset L_1$ , and the set $\mathcal {L}\setminus L_1$ has cardinality $\mathfrak {c}$ . Let $S= \{s_1, s_2, \ldots , s_n, \ldots \}$ be any countably infinite subset of $\mathcal {L}\setminus L_1$ . As $\mathcal {L}\setminus L_1$ has cardinality $\mathfrak {c}$ , there are $\mathfrak {c}$ distinct such subsets S. Then, $\mathcal {L}\setminus S= \bigcap _{i=1}^\infty (\mathcal {L}\setminus \{s_i\})$ .

Observing that $\mathcal {L}\supset \mathcal {L}\setminus S\supset L_1$ , Proposition 2.10 implies that each $\mathcal {L}\setminus S$ is an Erdős–Liouville set.

Noting that $\mathcal {L}$ is a $G_{\delta }$ -set in $\mathbb {R}$ , and each $\mathcal {L}\setminus \{s_i\}$ is an open set in $\mathcal {L}$ , it follows that $\mathcal {L}\setminus S$ is a $G_{\delta }$ -set. By Theorem 3.6, each of the $\mathfrak {c}$ sets $\mathcal {L}\setminus S$ is therefore homeomorphic to $ \mathcal {L}$ and $\mathbb {P}$ .