On the Hausdorff dimension of certain sets arising in Number Theory

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This paper has been withdrawn Any real number $x$ in the unit interval can be expressed as a continued fraction $x=[n_1,...,n_{{}_N},...]$.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 $\frac{{\sum }_{i=1}^N{n}_i}{N}\le m$.

arXiv commentWe add a reference and we correct the Hausdorff dimension of E_2