The arithmetic-geometric scaling spectrum for continued fractions

Details The arithmetic-geometric scaling spectrum for continued fractions The arithmetic-geometric scaling spectrum for continued fractions

2008-08-18

arxiv.org/abs/0808.2308v2

Arkiv f\"or Matematik 48 (2010), no. 2

Mathematics

Number Theory

22 pages, 1 figure

Scientific paper

10.1007/s11512-009-0102-8

To compare continued fraction digits with the denominators of the corresponding approximants we introduce the arithmetic-geometric scaling. We will completely determine its multifractal spectrum by means of a number theoretical free energy function and show that the Hausdorff dimension of sets consisting of irrationals with the same scaling exponent coincides with the Legendre transform of this free energy function. Furthermore, we identify the asymptotic of the local behaviour of the spectrum at the right boundary point and discuss a connection to the set of irrationals with continued fraction digits exceeding a given number which tends to infinity.

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