LIOUVILLE NUMBER GROUPS: STRUCTURE, CARDINALITY AND STRONG GENERATORS

Abstract We introduce and study Liouville number groups, namely additive subgroups of the real numbers in which every nonzero element is a Liouville number. Using continued-fraction methods and linear independence over the field of algebraic numbers, we establish the existence of large families of such groups with rich algebraic and topological structure. We prove that there exist <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow> <mml:mrow> <mml:mi mathvariant="fraktur">c</mml:mi> </mml:mrow> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:math> $(2^{\mathfrak {c}}) $ left parenthesis 2 Superscript German c Baseline right parenthesis pairwise distinct Liouville number groups generated by strong Liouville numbers and that, among these, there are <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow> <mml:mrow> <mml:mi mathvariant="fraktur">c</mml:mi> </mml:mrow> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:math> $(2^{\mathfrak {c}})$ left parenthesis 2 Superscript German c Baseline right parenthesis pairwise nonhomeomorphic groups. We further show that there exist continuum many countable Liouville number groups, each generated by countably many strong Liouville numbers and homeomorphic, as topological spaces, to the rational numbers. In addition, we prove that no subgroup of the real numbers is homeomorphic to the space of Liouville numbers, thereby highlighting a strong topological distinction between the space of Liouville numbers and Liouville number groups.