3 research outputs found
Normal Numbers and the Borel Hierarchy
We show that the set of absolutely normal numbers is -complete in the Borel hierarchy of subsets of real numbers. Similarly,
the set of absolutely normal numbers is -complete in the effective
Borel hierarchy
Borel complexity of sets of normal numbers via generic points in subshifts with specification
We study the Borel complexity of sets of normal numbers in several numeration
systems. Taking a dynamical point of view, we offer a unified treatment for
continued fraction expansions and base expansions, and their various
generalisations: generalised L\"uroth series expansions and -expansions.
In fact, we consider subshifts over a countable alphabet generated by all
possible expansions of numbers in . Then normal numbers correspond to
generic points of shift-invariant measures. It turns out that for these
subshifts the set of generic points for a shift-invariant probability measure
is precisely at the third level of the Borel hierarchy (it is a
-complete set, meaning that it is a countable intersection of
-sets, but it is not possible to write it as a countable union of
-sets). We also solve a problem of Sharkovsky--Sivak on the Borel
complexity of the basin of statistical attraction. The crucial dynamical
feature we need is a feeble form of specification. All expansions named above
generate subshifts with this property. Hence the sets of normal numbers under
consideration are -complete.Comment: A talk explaining this paper may be found at
https://www.youtube.com/watch?v=g9va0ZzVIj
Normal Numbers and the Borel Hierarchy
We show that the set of absolutely normal numbers is Π0 3-complete in the Borel hierarchy of subsets of real numbers. Similarly, the set of absolutely normal numbers is Π0 3-complete in the effective Borel hierarchy.