On the Hausdorff dimension of Furstenberg sets and orthogonal projections in the plane

Abstract

Let 0 s 1 and 0 t 2. An .s;t/-Furstenberg set is a set K R2 with the following property: there exists a line set L of Hausdorff dimension dimH L t such that dimH.K \ `/ s for all ` 2 L. We prove that for s 2 .0;1/ and t 2 .s;2, the Hausdorff dimension of .s;t/-Furstenberg sets in R2 is no smaller than 2s C , where >0 depends only on s and t. For s > 1=2 and t D 1, this is an -improvement over a result of Wolff from 1999. The same method also yields an -improvement to Kaufman’s projection theorem from 1968. We show that if s 2 .0;1/, t 2 .s;2, and K R2 is an analytic set with dimH K D t, then dimH ® e 2 S1 W dimH e.K/ s ¯ s ; where >0 depends only on s and t. Here e is the orthogonal projection to the line in direction e.