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

Let 0 ď s ď 1 and 0 ď t ď 2. An ps, tq-Furstenberg set is a setK Ă R with the following property: there exists a line set L of Hausdorff dimension dimH L ě t such that dimHpK X `q ě s for all ` P L. We prove that for s P p0, 1q, and t P ps, 2s, the Hausdorff dimension of ps, tq-Furstenberg sets in R is no smaller than 2s` , where ą 0 depends only on s and t. For s ą 1{2 and t “ 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 P p0, 1q, t P ps, 2s and K Ă R is an analytic set with dimH K “ t, then dimHte P S : dimH πepKq ď su ď s ́ , where ą 0 only depends on s and t. Here πe is the orthogonal projection to spanpeq.