On the Hausdorff dimension of certain sets arising in Number Theory

Eda Cesaratto

On the Hausdorff dimension of certain sets arising in Number Theory

Number Theory 2012-01-20 v3

Authors: Eda Cesaratto

Abstract

This paper has been withdrawn Any real number $x$ in the unit interval can be expressed as a continued fraction $x=[n_1,...,n_{_N},...]$ . Subsets of zero measure are obtained by imposing simple conditions on the $n_{_N}$ . By imposing $n_{_N}\le m \forall N\in \zN$ , Jarnik defined the corresponding sets $E_m$ and gave a first estimate of $d_H(E_m)$ , $d_H$ the Hausdorff dimension. Subsequent authors improved these estimates. In this paper we deal with $d_H(E_m)$ and $d_H(F_m)$ , $F_m$ being the set of real numbers for which ${\sum_{i=1}^N n_i\over N}\le m$ .

Keywords

hausdorff dimension continued fractions zero measure sets jarnik sets number theory

Cite

@article{arxiv.math/9908043,
  title  = {On the Hausdorff dimension of certain sets arising in Number Theory},
  author = {Eda Cesaratto},
  journal= {arXiv preprint arXiv:math/9908043},
  year   = {2012}
}

Comments

We add a reference and we correct the Hausdorff dimension of E_2

On the Hausdorff dimension of certain sets arising in Number Theory

Abstract

Keywords

Cite

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