226 research outputs found
On the Normality of Numbers to Different Bases
We prove independence of normality to different bases We show that the set of
real numbers that are normal to some base is Sigma^0_4 complete in the Borel
hierarchy of subsets of real numbers. This was an open problem, initiated by
Alexander Kechris, and conjectured by Ditzen 20 years ago
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
Irrationality exponent, Hausdorff dimension and effectivization
We generalize the classical theorem by Jarnik and Besicovitch on the irrationality exponents of real numbers and Hausdorff dimension and show that the two notions are independent. For any real number a greater than or equal to 2 and any non-negative real b be less than or equal to 2 / a, we show that there is a Cantor-like set with Hausdorff dimension equal to b such that, with respect to its uniform measure, almost all real numbers have irrationality exponent equal to a. We give an analogous result relating the irrationality exponent and the effective Hausdorff dimension of individual real numbers. We prove that there is a Cantor-like set such that, with respect to its uniform measure, almost all elements in the set have effective Hausdorff dimension equal to b and irrationality exponent equal to a. In each case, we obtain the desired set as a distinguished path in a tree of Cantor sets.Fil: Becher, Veronica Andrea. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigación en Ciencias de la Computación. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigación en Ciencias de la Computación; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; ArgentinaFil: Reimann, Jan. State University of Pennsylvania; Estados UnidosFil: Slaman, Theodore A.. University of California. Department of Mathematics; Estados Unido
The density of infima in the recursively enumerable degrees
AbstractWe show that every nontrivial interval in the recursively enumerable degrees contains an incomparable pair which have an infimum in the recursively enumerable degrees
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