Volume of the set of Liouville numbers #
In this file we prove that the set of Liouville numbers with exponent (irrationality measure)
strictly greater than two is a set of Lebesuge measure zero, see
Since this set is a residual set, we show that the filters
volume.ae are disjoint.
These filters correspond to two common notions of genericity on
ℝ: residual sets and sets of full
measure. The fact that the filters are disjoint means that two mutually exclusive properties can be
“generic” at the same time (in the sense of different “genericity” filters).
Liouville number, Lebesgue measure, residual, generic property
The set of numbers satisfying the Liouville condition with some exponent
p > 2 has Lebesgue
The set of Liouville numbers has Lebesgue measure zero.
residual ℝ and
volume.ae are disjoint. This means that there exists a residual
set of Lebesgue measure zero (e.g., the set of Liouville numbers).