Finding subsets of positive measure

An important theorem of geometric measure theory (first proved by Besicovitch and Davies for Euclidean space) says that every analytic set of non-zero $s$-dimensional Hausdorff measure $\mathcal H^s$ contains a closed subset of non-zero (and indeed finite) $\mathcal H^s$-measure. We investigate the question how hard it is to find such a set, in terms of the index set complexity, and in terms of the complexity of the parameter needed to define such a closed set. Among other results, we show that given a (lightface) $\Sigma^1_1$ set of reals in Cantor space, there is always a $\Pi^0_1(\mathcal{O})$ subset on non-zero $\mathcal H^s$-measure definable from Kleene's $\mathcal O$. On the other hand, there are $\Pi^0_2$ sets of reals where no hyperarithmetic real can define a closed subset of non-zero measure.Comment: This is an extended journal version of the conference paper "The Strength of the Besicovitch--Davies Theorem". The final publication of that paper is available at Springer via http://dx.doi.org/10.1007/978-3-642-13962-8_2

Independence, Relative Randomness, and PA Degrees

We study pairs of reals that are mutually Martin-L\"{o}f random with respect to a common, not necessarily computable probability measure. We show that a generalized version of van Lambalgen's Theorem holds for non-computable probability measures, too. We study, for a given real $A$, the \emph{independence spectrum} of $A$, the set of all $B$ so that there exists a probability measure $\mu$ so that $\mu\{A,B\} = 0$ and $(A,B)$ is $\mu\times\mu$-random. We prove that if $A$ is r.e., then no $\Delta^0_2$ set is in the independence spectrum of $A$. We obtain applications of this fact to PA degrees. In particular, we show that if $A$ is r.e.\ and $P$ is of PA degree so that $P \not\geq_{T} A$, then $A \oplus P \geq_{T} 0'$

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 Científicas 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

Particle sorting by a structured microfluidic ratchet device with tunable selectivity: Theory and Experiment

We theoretically predict and experimentally demonstrate that several different particle species can be separated from each other by means of a ratchet device, consisting of periodically arranged triangular (ratchet) shaped obstacles. We propose an explicit algorithm for suitably tailoring the externally applied, time-dependent voltage protocol so that one or several, arbitrarily selected particle species are forced to migrate oppositely to all the remaining species. As an example we present numerical simulations for a mixture of five species, labelled according to their increasing size, so that species 2 and 4 simultaneously move in one direction and species 1, 3, and 5 in the other. The selection of species to be separated from the others can be changed at any time by simply adapting the voltage protocol. This general theoretical concept to utilize one device for many different sorting tasks is experimentally confirmed for a mixture of three colloidal particle species

Turing Degrees and Randomness for Continuous Measures

We study degree-theoretic properties of reals that are not random with respect to any continuous probability measure (NCR). To this end, we introduce a family of generalized Hausdorff measures based on the iterates of the "dissipation" function of a continuous measure and study the effective nullsets given by the corresponding Solovay tests. We introduce two constructions that preserve non-randomness with respect to a given continuous measure. This enables us to prove the existence of NCR reals in a number of Turing degrees. In particular, we show that every $\Delta^0_2$-degree contains an NCR element.Comment: 22 page