There are two fundamental results in the classical theory of metric Diophantine approximation: Khintchine’s theorem and Jarńık’s theorem. The former relates the size of the set of well approximable numbers, expressed in terms of Lebesgue measure, to the behavior of a certain volume sum. The latter is a Hausdorff measure version of the former. We start by discussing these theorems and show that ...

It turns out that H1 is a measure, now called one-dimensional Hausdorff measure because it was generalized by Hausdorff [10] to the whole family of measures Hα, where α is any positive number (integer or noninteger). The modern theory of “fractals” is largely based on the notion of the Hausdorff dimension dimH (F) of a set F , defined by dimH (F) = inf{α > 0 : Hα(F) = 0}. We recommend the book ...

The notion of compatible apparition points is introduced for non-Hausdorff manifolds, and properties of these points are studied. It is well known that the Hausdorff property is independent of the other conditions given in the standard definition of a topological manifold. In much of literature, a topological manifold of dimension n is a Hausdorff topological space which has a countable base of...

The Hausdorff dimension of a product X × Y can be strictly greater than that of Y , even when the Hausdorff dimension of X is zero. But when X is countable, the Hausdorff dimensions of Y and X × Y are the same. Diagonalizations of covers define a natural hierarchy of properties which are weaker than “being countable” and stronger than “having Hausdorff dimension zero”. Fremlin asked whether it ...

Hausdorff dimension assigns a dimension value to each subset of an arbitrary metric space. In Euclidean space, this concept coincides with our intuition that smooth curves have dimension 1 and smooth surfaces have dimension 2, but from its introduction in 1918 [23] Hausdorff noted that many sets have noninteger dimension, what he called “fractional dimension”. The development and applications o...

The existence and Hausdorff dimension of the global attractor for discretization of a damped wave equation with the periodic nonlinearity under the periodic boundary conditions are studied for any space dimension. The obtained Hausdorff dimension is independent of the mesh sizes and the space dimension and remains small for large damping, which conforms to the physics.

A very important property of a deterministic self-similar set is that its Hausdorff dimension and upper box-counting dimension coincide. This paper considers the random case. We show that for a random self-similar set, its Hausdorff dimension and upper box-counting dimension are equal a.s.

In a recent paper, Pertti Mattila asked which gauge functions φ have the property that for any Borel set A ⊂ R2 with Hausdorff measure Hφ(A) > 0, the projection of A to almost every line has positive length. We show that finiteness of ∫ 1 0 φ(r) r2 dr, which is known to be sufficient for this property, is also necessary for regularly varying φ. Our proof is based on a random construction adapte...

We study the Hausdorff dimension and the pointwise dimension of measures that are not necessarily ergodic. In particular, for conformal expanding maps and hyperbolic diffeomorphisms on surfaces we establish explicit formulas for the pointwise dimension of an arbitrary invariant measure in terms of the local entropy and of the Lyapunov exponents. These formulas are obtained with a direct approac...

In the present survey paper, we explain how the theory of Hausdorff dimension and Hausdorff measure is used to answer certain questions in Diophantine approximation. The final section is devoted to a discussion around the Diophantine properties of the points lying in the middle third Cantor set.