2010/06/24 by Ursula Molter, Molter, Ursula, Ezequiel Rela +1
Computer Science · Mathematics · #28A78 #28A80 #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #Digital Image Processing Techniques #FOS: Mathematics #Mathematical Approximation and Integration #math.CA #math.CO #msc:28A78 #msc:28A80
paper · pdf · doi:10.48550/arxiv.1006.4862
Final version
openalex publication_date 2010/06/24 · arxiv created 2012/11/12 · arxiv updated 2012/11/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For α in (0,1], a subset E of \RR is called Furstenberg set of type α or Fα-set if for each direction e in the unit circle there is a line segment ℓe in the direction of e such that the Hausdorff dimension of the set E∩ℓe is greater or equal than α. In this paper we show that if α> 0, there exists a set E∈ Fα such that \HHg(E)=0 for g(x)=x1/2+3/2αlog-θ((1)/(x)), θ>(1+3α)/(2), which improves on the the previously known bound, that Hβ(E) = 0 for β>1/2+3/2α. Further, by refining the argument in a subtle way, we are able to obtain a sharp dimension estimate for a whole class of zero-dimensional Furstenberg type sets. Namely, for \hγ(x)=log-γ((1)/(x)), γ>0, we construct a set Eγ∈ F\hγ of Hausdorff dimension not greater than 1/2. Since in a previous work we showed that 1/2 is a lower bound for the Hausdorff dimension of any E∈ F\hγ, with the present construction, the value 1/2 is sharp for the whole class of Furstenberg sets associated to the zero dimensional functions \hγ.