ERDŐS PROPERTIES OF SUBSETS OF THE MAHLER SETS

Abstract

AbstractErdős proved that every real number is the sum of two Liouville numbers. A setWof complex numbers is said to have the Erdős property if every real number is the sum of two members ofW. Mahler divided the set of all transcendental numbers into three disjoint classesS,TandUsuch 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 setUand 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 subsetsWofSeach of Lebesgue measuremsuch thatWhas the Erdős property and no two of theseWare homeomorphic. It is also proved that there are$2^{\mathfrak {c}}$dense subsetsWofSeach of full Lebesgue measure, which have the Erdős property. Finally, it is proved that there are$2^{\mathfrak {c}}$dense subsetsWofSsuch that every complex number is the sum of two members ofWand such that no two of theseWare homeomorphic.