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Refined size estimates for Furstenberg sets via Hausdorff measures: a survey of some recent results

In this survey we collect and discuss some recent results on the so called "Furstenberg set problem", which in its classical form concerns the estimates of the Hausdorff dimension of planar sets containing, for any direction, a subset of an interval poitning in that direction of some prescribed dimension. This problem is closely related to the "Kakeya needle problem". In this work we approach this problem from a more general point of view, in terms of generalized Hausdorff measures associated to dimension functions. We generalize the known results in terms of "logarithmic gaps" and obtain analogues to the classical estimates. Moreover, these analogues allow us to extend our results to the zero dimensional endpoint. We also obtain results about the dimension of a variation of Furstenberg sets defined for a fractal set of directions. We prove analogous inequalities reflecting the interplay between the size of the set of directions and the size of each fiber. This problem is also studied in the general scenario of Hausdorff measures.