60,273 research outputs found
Finite-State Dimension and Real Arithmetic
We use entropy rates and Schur concavity to prove that, for every integer k
>= 2, every nonzero rational number q, and every real number alpha, the base-k
expansions of alpha, q+alpha, and q*alpha all have the same finite-state
dimension and the same finite-state strong dimension. This extends, and gives a
new proof of, Wall's 1949 theorem stating that the sum or product of a nonzero
rational number and a Borel normal number is always Borel normal.Comment: 15 page
p-adic path set fractals and arithmetic
This paper considers a class C(Z_p) of closed sets of the p-adic integers
obtained by graph-directed constructions analogous to those of Mauldin and
Williams over the real numbers. These sets are characterized as collections of
those p-adic integers whose p-adic expansions are describeed by paths in the
graph of a finite automaton issuing from a distinguished initial vertex. This
paper shows that this class of sets is closed under the arithmetic operations
of addition and multiplication by p-integral rational numbers. In addition the
Minkowski sum (under p-adic addition) of two set in the class is shown to also
belong to this class. These results represent purely p-adic phenomena in that
analogous closure properties do not hold over the real numbers. We also show
the existence of computable formulas for the Hausdorff dimensions of such sets.Comment: v1 24 pages; v2 added to title, 28 pages; v3, 30 pages, added
concluding section, v.4, incorporate changes requested by reviewe
Totally Geodesic Spectra of Arithmetic Hyperbolic Spaces
In this paper we show that totally geodesic subspaces determine the
commensurability class of a standard arithmetic hyperbolic -orbifold, . Many of the results are more general and apply to locally symmetric spaces
associated to arithmetic lattices in -simple Lie groups of type
and . We use a combination of techniques from algebraic groups and
quadratic forms to prove several results about these spaces.Comment: 34 Pages. Corrected typos. Added references. Improved expositio
- β¦