Mean dimension and Jaworski-type theorems

Plünnecke and Kneser type theorems for dimension estimates

Given a division ring K containing the field k in its center and two finite subsets A and B of K∗, we give some analogues of Plünnecke and Kneser Theorems for the dimension of the k-linear span of the Minkowski product AB in terms of the dimensions of the k-linear spans of A and B. We also explain how they imply the corresponding more classical theorems for abelian groups. These Plünnecke type ...

Hausdorff Dimension and mean porosity

Intersection theorems under dimension constraints

In [8] we posed a series of extremal (set system) problems under dimension constraints. In the present paper we study one of them: the intersection problem. The geometrical formulation of our problem is as follows. Given integers 0 ≤ t, k ≤ n determine or estimate the maximum number of (0, 1)–vectors in a k–dimensional subspace of the Euclidean n–space Rn, such that the inner product (“intersec...

Projection Theorems Using Effective Dimension

In this paper we use the theory of computing to study fractal dimensions of projections in Euclidean spaces. A fundamental result in fractal geometry is Marstrands projection theorem, which shows that for every analytic set E, for almost every line L, the Hausdorff dimension of the orthogonal projection of E onto L is maximal. We use Kolmogorov complexity to give two new results on the Hausdorf...

Sofic Mean Dimension

We introduce mean dimensions for continuous actions of countable sofic groups on compact metrizable spaces. These generalize the Gromov-LindenstraussWeiss mean dimensions for actions of countable amenable groups, and are useful for distinguishing continuous actions of countable sofic groups with infinite entropy.