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 dimHL≥t such that dimH(K∩ℓ)≥s for all ℓ∈L. We prove that for s∈(0,1) and t∈(s,2], the Hausdorff dimension of (s,t)-Furstenberg sets in R2 is no smaller than 2s+ϵ, where ϵ>0 depends only on s and t. For s>1∕2 and t=1, this is an ϵ-improvement over a result of Wolff from 1999. The same method also yields an ϵ-improvement to Kaufman’s projection theorem from 1968. We show that if s∈(0,1), t∈(s,2], and K⊂R2 is an analytic set with dimHK=t, then dimH{e∈S1:dimHπe(K)≤s}≤s−ϵ, where ϵ>0 depends only on s and t. Here πe is the orthogonal projection to the line in direction e.

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