TY - JOUR

T1 - Irrationality exponent, Hausdorff dimension and effectivization

AU - Becher, Verónica

AU - Reimann, Jan

AU - Slaman, Theodore A.

N1 - Funding Information:
V. Becher’s research is supported by the University of Buenos Aires and CONICET. V. Becher is member of the Laboratoire International Associé INFINIS, CONICET/Universidad de Buenos Aires-CNRS/Université Paris Diderot. J. Reimann was partially supported by NSF Grant DMS-1201263. T. A. Slaman was partially supported by NSF Grant DMS-1600441.

PY - 2018/2/1

Y1 - 2018/2/1

N2 - 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.

AB - 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.

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U2 - 10.1007/s00605-017-1094-2

DO - 10.1007/s00605-017-1094-2

M3 - Article

AN - SCOPUS:85029083035

VL - 185

SP - 167

EP - 188

JO - Monatshefte fur Mathematik

JF - Monatshefte fur Mathematik

SN - 0026-9255

IS - 2

ER -