2008/07/18 by Michael Bateman, Alexander Volberg, Bateman, Michael +1 · 2 citations
Computer Science · Mathematics · #28A80 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Mathematical Dynamics and Fractals #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.0807.2953
openalex publication_date 2008/07/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \Cantn be the n-th generation in the construction of the middle-half Cantor set. The Cartesian square \Kn = \Cantn × \Cantn consists of 4n squares of side-length 4-n. The chance that a long needle thrown at random in the unit square will meet \Kn is essentially the average length of the projections of \Kn, also known as the Favard length of \Kn. A classical theorem of Besicovitch implies that the Favard length of \Kn tends to zero. It is still an open problem to determine its exact rate of decay. Until recently, the only explicit upper bound was exp(- clog_* n), due to Peres and Solomyak. (log_* n is the number of times one needs to take log to obtain a number less than 1 starting from n). In Nazarov-Peres-Volberg paper (arxiv:math 0801.2942) the power estimate from above was obtained. The exponent in this paper was less than 1/6 but could have been slightly improved. On the other hand, a simple estimate shows that from below we have the estimate (c)/(n). Here we apply the idea from papers of Nets Katz (MRL (1996), 527-536) and Bateman-Katz (arxiv:math/0609187v1 2006) to show that the estimate from below can be in fact improved to c (log n)/(n). This is in drastic difference from the case of \em random Cantor sets studied by Peres and Solomyak in Pacific J. Math. 204 (2002), 473-496.