58,623 research outputs found

    Finite-State Dimension and Real Arithmetic

    Get PDF
    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

    Full text link
    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

    Full text link
    In this paper we show that totally geodesic subspaces determine the commensurability class of a standard arithmetic hyperbolic nn-orbifold, nβ‰₯4n\ge 4. Many of the results are more general and apply to locally symmetric spaces associated to arithmetic lattices in R\mathbb{R}-simple Lie groups of type BnB_n and DnD_n. 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
    • …
    corecore