Improving dimension estimates for Furstenberg-type sets

Bit-size estimates for triangular sets in positive dimension

We give bit-size estimates for the coefficients appearing in triangular sets describing positive-dimensional algebraic sets defined over Q. These estimates are worst case upper bounds; they depend only on the degree and height of the underlying algebraic sets. We illustrate the use of these results in the context of a modular algorithm. This extends results by the first and last author, which w...

Furstenberg Sets for a Fractal Set of Directions

In this paper we study the behavior of the size of Furstenberg sets with respect to the size of the set of directions defining it. For any pair α, β ∈ (0, 1], we will say that a set E ⊂ R2 is an Fαβ-set if there is a subset L of the unit circle of Hausdorff dimension at least β and, for each direction e in L, there is a line segment e in the direction of e such that the Hausdorff dimension of t...

On estimates of Hausdorff dimension of invariant compact sets

In this paper we present two approaches to estimate the Hausdorff dimension of an invariant compact set of a dynamical system: the method of characteristic exponents (estimates of KaplanYorke type) and the method of Lyapunov functions. In the first approach, using Lyapunov's first method we exploit charachteristic exponents for obtaining such estimate. A close relationship with uniform asymptot...

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 ...

Dimension Estimates for Invariant Setsof