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Register a new userHausdorff dimension of the set of real numbers of Lagrange value three
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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.
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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 These results improve upon previous estimates of R. Baker, joint work of the first author with M. Laurent, and unpublished work of M. Laurent.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. .. .
This paper has been withdrawn Any real number $x$ in the unit interval can be expressed as a continued fraction $x=[n_1,...,n_{_N},...]$.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 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.
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.
It was discovered some years ago that there exist non-integer real numbers $q>1$ for which only one sequence $(c_i)$ of integers $c_i in [0,q)$ satisfies the equality $sum_{i=1}^infty c_iq^{-i}=1$. The set of such univoque numbers has a rich topological structure, and its study revealed a number of unexpected connections with measure theory, fractals, ergodic theory and Diophantine approximation. In this paper we consider for each fixed $q>1$ the set $mathcal{U}_q$ of real numbers $x$ having a unique representation of the form $sum_{i=1}^infty c_iq^{-i}=x$ with integers $c_i$ belonging to $[0,q)$. We carry out a detailed topological study of these sets. For instance, we characterize their closures, and we determine those bases $q$ for which $mathcal{U}_q$ is closed or even a Cantor set. We also study the set $mathcal{U}_q$ consisting of all sequences $(c_i)$ of integers $c_i in [0,q)$ such that $sum_{i=1}^{infty} c_i q^{-i} in mathcal{U}_q$. We determine the numbers $r >1$ for which the map $q mapsto mathcal{U}_q$ (defined on $(1, infty)$) is constant in a neighborhood of $r$ and the numbers $q >1$ for which $mathcal{U}_q$ is a subshift or a subshift of finite type.
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