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 
$\mathcal {L}$
 .


Introduction
It has been known for over 175 years that every Liouville number is transcendental and for 120 years that the set L of Liouville numbers is uncountable. Notwithstanding this, the set L is known to have Lebesgue measure zero. So in this sense, 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 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 [2] Erdős-Liouville sets 285 than L which have this property. Indeed, there are 2 c such subsets of L no two of which are homeomorphic as subspaces of R.

Preliminaries
REMARK 2.1. In 1844, Joseph Liouville proved the existence of transcendental numbers [2,3]. He introduced the set 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 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 ∈ N, there are infinitely many k such that q k+1 > q n k . A more restrictive class of Liouville numbers is obtained by requiring this inequality to hold for every k > N = N(n) ∈ 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 L of all Liouville numbers is a dense G δ -set in R and showed that every dense G δ -set in R has this property.   [4,13].) REMARK 2.4. By the theorem proved by Erdős mentioned above, the set 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 L of Liouville numbers. REMARK 2.7. It is not immediately obvious that there exist any Erdős-Liouville sets other than the set L itself. It is known that some sets of positive Lebesgue measure have the Erdős property, but they are not subsets of L as the set 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 δ -subset of 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 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 c dense G δ subsets of R. The relevant theorem from [9] describing this stronger property is the following result. If we put f (x) = r − x, for r, x ∈ R and I = 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, √ r) and f (x) = √ 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(α) : α ∈ L} leads to the observation that L 1 ∩ 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 ∩ L, for n ∈ 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 L. PROPOSITION 2.10. Let S be a set of real numbers such that W 1 ⊃ S ⊃ W 2 , where W 1 and W 2 are Erdős-Liouville sets. Then S is an Erdős-Liouville set.
PROOF. As S ⊃ W 2 , by Remark 2.5, it has the Erdős property. Also as W 2 is dense in R, so too is S. Finally, as S ⊂ W 1 , it is a subset of L. Therefore, S is an Erdős-Liouville set.

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 P the set of all irrational real numbers with the topology it inherits as a subspace of the euclidean space R.
A beautiful characterisation of the topological space P is given in [16, Theorem 1.9.8].
THEOREM 3.2. The space of all irrational real numbers P is topologically the unique nonempty, separable, metrisable, topologically complete, nowhere locally compact, and zero-dimensional space.
This has a Corollary 3.3, [16, Corollary 1.9.9], which is often proved using continued fractions. COROLLARY 3.3. The space P is homeomorphic to the Tychonoff product N ℵ 0 of a countably infinite number of homeomorphic copies of the discrete space N of positive integers. Hence, P × P is homeomorphic to P. Indeed, P is homeomorphic to P ℵ 0 . REMARK 3.4. Recall that a subset X of a topological space Y is said to be a G δ -set if it is a countable intersection of open sets in Y while X is said to be an F σ -set if it is a countable union of closed sets in Y. Obviously, a subset X of a topological space Y is a G δ -set if and only if its complement is an F σ -set. We see immediately that in a metric space such as R, the set T of all transcendental real numbers is a G δ -set as its complement is the countably infinite set A of all real algebraic numbers. Now we connect the notion of G δ -set in 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 δ -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 δ subset of the set P of all irrational real numbers is homeomorphic to P and to N ℵ 0 . In particular, the space T of all real transcendental numbers and the space L of all Liouville numbers, with their subspace topologies from R, are both homeomorphic to P and to N ℵ 0 .
These results and a similar one [16, Theorem 1.9.6] characterising the space 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 c Erdős-Liouville sets THEOREM 4.1. Let X be a topological space homeomorphic to P. Then X has a dense G δ -set Y which is homeomorphic to P such that the cardinality of the set X \ Y is c, the cardinality of the continuum.
PROOF. Consider the topological space T of all real transcendental numbers and the topological space L of all Liouville numbers. We saw in Corollary 2.6 and Remark 2.1 that L is a dense G δ -set, and T and L are homeomorphic to P. Further, the cardinality of the set T \ L is c. As the properties of being a dense G δ -set and having cardinality c are preserved by homeomorphisms, the theorem is proved.
By Theorem 4.1 and Remark 2.1, we have the following corollary.
with each L n \ L n+1 having cardinality c and each L n+1 a G δ -set in L n which is homeomorphic to P. PROOF. First, we note that there are precisely 2 c subsets of the set L of all Liouville numbers as L has cardinality c. So the cardinality of the set of Erdős-Liouville sets is not greater than 2 c .
Using the notation of Theorem 4.4, let W be any subset of L \ L 1 . As L 1 is an Erdős-Liouville set and L 1 ⊂ L, Remark 2.5 implies that L 1 ∪ W is an Erdős-Liouville set. As there are 2 c subsets W of the set L \ L 1 , it follows that there are 2 c distinct Erdős-Liouville sets. So it remains to show only that amongst these, there are 2 c no two of which are homeomorphic.
By the Laverentieff theorem, [16,Theorem A8.5], there are at most c subspaces of R which are homeomorphic. As there are 2 c distinct Erdős-Liouville sets, it follows that there are 2 c Erdős-Liouville sets no two of which are homeomorphic, as required. THEOREM 4.6. There exist c Erdős-Liouville sets each of which is homeomorphic to L with its subspace topology. So each is homeomorphic to P. PROOF. Using the notation of Theorem 4.4, L ⊃ L 1 , and the set L \ L 1 has cardinality c. Let S = {s 1 , s 2 , . . . , s n , . . .} be any countably infinite subset of L \ L 1 . As L \ L 1 has cardinality c, there are c distinct such subsets S. Then, L \ S = ∞ i=1 (L \ {s i }). Observing that L ⊃ L \ S ⊃ L 1 , Proposition 2.10 implies that each L \ S is an Erdős-Liouville set.
Noting that L is a G δ -set in R, and each L \ {s i } is an open set in L, it follows that L \ S is a G δ -set. By Theorem 3.6, each of the c sets L \ S is therefore homeomorphic to L and P.