Search

Erdős proved that every real number is the sum of two Liouville numbers. A set W of complex numbers is said to have the Erdős property if every real number is the sum of two members of W. Mahler divided the set of all transcendental numbers into three disjoint classes S, T and U such that, in particular, any two complex numbers which are algebraically dependent lie in the same class. The set of Liouville numbers is a proper subset of the set U and has Lebesgue measure zero. It is proved here, using a theorem of Weil on locally compact groups, that if $m\in [0,\infty )$ , then there exist $2^{\mathfrak {c}}$ dense subsets W of S each of Lebesgue measure m such that W has the Erdős property and no two of these W are homeomorphic. It is also proved that there are $2^{\mathfrak {c}}$ dense subsets W of S each of full Lebesgue measure, which have the Erdős property. Finally, it is proved that there are $2^{\mathfrak {c}}$ dense subsets W of S such that every complex number is the sum of two members of W and such that no two of these W are homeomorphic.

For any Liouville number $\alpha $ , all of the following are transcendental numbers: ${e}^\alpha $ , $\log _{e}\alpha $ , $\sin \alpha $ , $\cos \alpha $ , $\tan \alpha $ , $\sinh \alpha $ , $\cosh \alpha $ , $\tanh \alpha $ , $\arcsin \alpha $ and the inverse functions evaluated at $\alpha $ of the listed trigonometric and hyperbolic functions, noting that wherever multiple values are involved, every such value is transcendental. This remains true if ‘Liouville number’ is replaced by ‘U-number’, where U is one of Mahler’s classes of transcendental numbers.

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 $\mathcal {L}$ .

A thin set is defined to be an uncountable dense zero-dimensional subset of measure zero and Hausdorff measure zero of an Euclidean space. A fat set is defined to be an uncountable dense path-connected subset of an Euclidean space which has full measure, that is, its complement has measure zero. While there are well-known pathological maps of a set of measure zero, such as the Cantor set, onto an interval, we show that the standard addition on $\mathbb {R}$ maps a thin set onto a fat set; in fact the fat set is all of $\mathbb {R}$ . Our argument depends on the theorem of Paul Erdős that every real number is a sum of two Liouville numbers. Our thin set is the set $\mathcal {L}^{2}$ , where $\mathcal {L}$ is the set of all Liouville numbers, and the fat set is $\mathbb {R}$ itself. Finally, it is shown that $\mathcal {L}$ and $\mathcal {L}^{2}$ are both homeomorphic to $\mathbb {P}$ , the space of all irrational numbers.

This article initiates the study of topological transcendental fields F which are subfields of the topological field C of all complex numbers such that F only consists of rational numbers and a nonempty set of transcendental numbers. F, with the topology it inherits as a subspace of C, is a topological field. Each topological transcendental field is a separable metrizable zero-dimensional space and algebraically is Q(T), the extension of the field of rational numbers by a set T of transcendental numbers. It is proven that there exist precisely 2ℵ0 countably infinite topological transcendental fields and each is homeomorphic to the space Q of rational numbers with its usual topology. It is also shown that there is a class of 22ℵ0 of topological transcendental fields of the form Q(T) with T a set of Liouville numbers, no two of which are homeomorphic.