We establish formulas for bounds on the Haudorff measure of the intersection of certain Cantor sets with their translates. As a consequence we obtain a formula for the Hausdorff dimensions of these intersections.

Using Voiculescu’s notion of a matricial microstate we introduce fractal dimensions and entropies for finite sets of selfadjoint operators in a tracial von Neumann algebra. We show that they possess properties similar to their classical predecessors. We relate the new quantities to free entropy and free entropy dimension and show that a modified version of free Hausdorff dimension is an algebra...

Following Lutz’s approach to effective (constructive) dimension, we define a notion of dimension for individual sequences based on Schnorr’s concept(s) of randomness. In contrast to computable randomness and Schnorr randomness, the dimension concepts defined via computable martingales and Schnorr tests coincide, i.e. the Schnorr Hausdorff dimension of a sequence always equals its computable Hau...

It is known that the box dimension of any Martin-Löf random closed set of {0, 1}N is log2( 43 ). Barmpalias et al. [Journal of Logic and Computation, Vol. 17, No. 6 (2007)] gave one method of producing such random closed sets and then computed the box dimension, and posed several questions regarding other methods of construction. We outline a method using random recursive constructions for comp...

This paper studies labeled sample compression for multi-label concept classes. For a specific extension of the notion of VC-dimension to multi-label classes, we prove that every maximum multilabel class of dimension d has a sample compression scheme in which every sample is compressed to a subset of size at most d. We further show that every multi-label class of dimension 1 has a sample compres...

Let A, B, be finite subsets of an abelian group, and let G ⊂ A×B be such that #A,#B,#{a + b : (a, b) ∈ G} ≤ N . We consider the question of estimating the quantity #{a − b : (a, b) ∈ G}. In [2] Bourgain improved the trivial upper bound of N to N 1 13 , and applied this to the Kakeya conjecture. We improve Bourgain’s estimate further to N 1 6 , and conclude that Besicovitch sets in Rn have Hausd...

A relatively longstanding question in algorithmic randomness is Jan Reimann’s question whether there is a Turing cone of broken dimension. That is, is there a real A such that {B : B ≤T A} contains no 1-random real, yet contains elements of nonzero effective Hausdorff Dimension? We show that the answer is affirmative if Hausdorff dimension is replaced by its inner analogue packing dimension. We...