Improving dimension estimates for Furstenberg-type sets

MOLTER, URSULA; RELA, EZEQUIEL

ADVANCES IN MATHEMATICS

ACADEMIC PRESS INC ELSEVIER SCIENCE

Año: 2010 vol. 223 p. 672 - 688

0001-8708

In this paper we study the problem of estimating the generalized Hausdorff dimension of Furstenberg sets in the plane. For $alphain(0,1]$, a set $F$ in the plane is said to be an $alpha$-Furstenberg set if for each direction $e$ there is a line segment $ell_e$ in the direction of $e$ for which $dim_H(ell_ecap F)gealpha$. It is well known that $dim_H(F) geq max{2alpha, alpha + rac12}$ - and it is also known that these sets can have zero measure at their critical dimension. By looking at general Hausdorff measures $mathcal{H}^h$ defined for doubling functions, that need not be power laws, we obtain finer estimates for the size of the more general $h$-Furstenberg sets. Further, this approach allow us to sharpen the known bounds on the dimension of classical Furstenberg sets. The main difficulty we had to overcome, was that if $mathcal H^{h}(F) = 0$, there {em always} exists $g prec h$ such that $mathcal H^g(F) = 0$ (here $prec$ refers to the natural ordering on general Hausdorff dimension functions). Hence, in order to estimate the measure of general Furstenberg sets, we have to consider dimension functions that are a true {em step down} from the critical one. We provide rather precise estimates on the size of this step and by doing so, we can include a family of zero dimensional Furstenberg sets associated to dimension functions that grow faster than any power function at zero. With some additional growth conditions on these zero dimensional functions, we extend the known inequalities to include the endpoint $alpha=0$.