The traveling salesman problem is a problem of finding the shortest tour through given points. In 1959, Beardwood, Halton, and Hammersley studied asymptotic length of the shortest tour through points on Euclidean space. In particular they described the optimal tour length with Euclidean dimension for the case that points are distributed with respect to Lebesgue absolutely continuous measure. In...

In this paper we study the Hausdorff volume in a non equiregular sub-Riemannian manifold and we compare it with a smooth volume. We first give the Lebesgue decomposition of the Hausdorff volume. Then we study the regular part, show that it is not commensurable with the smooth volume, and give conditions under which it is a Radon measure. We finally give a complete characterization of the singul...

Lutz [7] has recently developed a constructive version of Hausdorff dimension, using it to assign to every sequence A ∈ C a constructive dimension dim(A) ∈ [0,1]. Classical Hausdorff dimension [3] is an augmentation of Lebesgue measure, and in the same way constructive dimension augments Martin– Löf randomness. All Martin–Löf random sequences have constructive dimension 1, while in the case of ...

We continue the study of effective Hausdorff dimension as it was initiated by LUTZ. Whereas he uses a generalization of martingales on the Cantor space to introduce this notion we give a characterization in terms of effective s-dimensional Hausdorff measures, similar to the effectivization of Lebesgue measure by MARTIN-LÖF. It turns out that effective Hausdorff dimension allows to classify sequ...

We construct Feigenbaum quadratic-like maps with a Julia set of positive Lebesgue measure. Indeed, in the quadratic family $P_c: z \mapsto z^2+c$ corresponding parameters $c$ is shown to have Hausdorff dimension. Our examples include renormalization fixed points, and polynomials their stable manifold are first known rational for which hyperbolic dimension different from set.