32 research outputs found
On the Hausdorff dimension of countable intersections of certain sets of normal numbers
We show that the set of numbers that are -distribution normal but not
simply -ratio normal has full Hausdorff dimension. It is further shown under
some conditions that countable intersections of sets of this form still have
full Hausdorff dimension even though they are not winning sets (in the sense of
W. Schmidt). As a consequence of this, we construct many explicit examples of
numbers that are simultaneously distribution normal but not simply ratio normal
with respect to certain countable families of basic sequences. Additionally, we
prove that some related sets are either winning sets or sets of the first
category.Comment: 12 pages, 1 figur
Unexpected distribution phenomenon resulting from Cantor series expansions
We explore in depth the number theoretic and statistical properties of
certain sets of numbers arising from their Cantor series expansions. As a
direct consequence of our main theorem we deduce numerous new results as well
as strengthen known ones.Comment: 32 page
On the transcendence of certain real numbers
In this article we discuss the transcendence of certain infinite sums and
products by using the Subspace theorem. In particular we improve the result of
Han\v{c}l and Rucki \cite{hancl3}.Comment: 14 page
Normal equivalencies for eventually periodic basic sequences
W. M. Schmidt, A. D. Pollington, and F. Schweiger have studied when normality
with respect to one expansion is equivalent to normality with respect to
another expansion. Following in their footsteps, we show that when is an
eventually periodic basic sequence, that -normality and -distribution
normality are equivalent to normality in base where is dependent on
. We also show that boundedness of the basic sequence is not sufficient for
this equivalence.Comment: 7 page