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Register a new userIrrationality Exponent, Hausdorff Dimension and Effectivization
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We generalize the classical theorem by Jarnik and Besicovitch on the irrationality exponents of real numbers and Hausdorff dimension. Let a be any real number greater than or equal to 2 and let b be any non-negative real 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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In this paper we prove the Hausdorff dimension of the set of (nondegenerate) singular two-dimensional vectors with uniform exponent $mu$ $in$ (1/2, 1) is 2(1 -- $mu$) when $mu$ $ge$ $sqrt$ 2/2, whereas for $mu$ textless{} $sqrt$ 2/2 it is greater than 2(1 -- $mu$) and at most (3 -- 2$mu$)(1 -- $mu$)/(1 + $mu$ + $mu$ 2). We also establish that this dimension tends to 4/3 (which is the dimension of the set of singular two-dimensional vectors) when $mu$ tends to 1/2. These results improve upon previous estimates of R. Baker, joint work of the first author with M. Laurent, and unpublished work of M. Laurent. We also prove a lower bound on the packing dimension that is strictly greater than the Hausdorff dimension for $mu$ $ge$ 0.565. .. .
We obtain the exact value of the Hausdorff dimension of the set of coefficients of Gauss sums which for a given $alpha in (1/2,1)$ achieve the order at least $N^{alpha}$ for infinitely many sum lengths $N$. For Weyl sums with polynomials of degree $dge 3$ we obtain a new upper bound on the Hausdorff dimension of the set of polynomial coefficients corresponding to large values of Weyl sums. Our methods also work for monomial sums, match the previously known lower bounds, just giving exact value for the corresponding Hausdorff dimension when $alpha$ is close to $1$. We also obtain a nearly tight bound in a similar question with arbitrary integer sequences of polynomial growth.
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 $n_{_N}le m forall Nin zN$, 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 ${sum_{i=1}^N n_iover N}le m$.
We show that the set of real numbers of Lagrange value 3 has Hausdorff dimension zero by showing the appropriate generalization for each element of the Teichmueller space of the appropriate subgroup of the classical modular group.
In [14], the authors developed a new approach to the computation of the Hausdorff dimension of the invariant set of an iterated function system or IFS. In this paper, we extend this approach to incorporate high order approximation methods. We again rely on the fact that we can associate to the IFS a parametrized family of positive, linear, Perron-Frobenius operators $L_s$, an idea known in varying degrees of generality for many years. Although $L_s$ is not compact in the setting we consider, it possesses a strictly positive $C^m$ eigenfunction $v_s$ with eigenvalue $R(L_s)$ for arbitrary $m$ and all other points $z$ in the spectrum of $L_s$ satisfy $|z| le b$ for some constant $b < R(L_s)$. Under appropriate assumptions on the IFS, the Hausdorff dimension of the invariant set of the IFS is the value $s=s_*$ for which $R(L_s) =1$. This eigenvalue problem is then approximated by a collocation method at the extended Chebyshev points of each subinterval using continuous piecewise polynomials of arbitrary degree $r$. Using an extension of the Perron theory of positive matrices to matrices that map a cone $K$ to its interior and explicit a priori bounds on the derivatives of the strictly positive eigenfunction $v_s$, we give rigorous upper and lower bounds for the Hausdorff dimension $s_*$, and these bounds converge rapidly to $s_*$ as the mesh size decreases and/or the polynomial degree increases.
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