Buffon's needle estimates for rational product Cantor sets

2011/09/05 by Bond, Matthew, Laba, Izabella, Volberg, Alexander · 3 citations
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1109.1031

Abstract

Let S_∞=A_∞× B_∞ be a self-similar product Cantor set in the complex plane, defined via S_∞=\bigcup_j=1^L T_j(S_∞), where T_j:\C→\C have the form T_j(z)=\frac1Lz+z_j and \z₁,...,z_L\=A+iB for some A,B⊂\rr with |A|,|B|>1 and |A||B|=L. Let S_N be the L^-N-neighbourhood of S_∞, or equivalently (up to constants), its N-th Cantor iteration. We are interested in the asymptotic behaviour as N→∞ of the \it Favard length of S_N, defined as the average (with respect to direction) length of its 1-dimensional projections. If the sets A and B are rational and have cardinalities at most 6, then the Favard length of S_N is bounded from above by CN^{-p/loglog N} for some p>0. The same result holds with no restrictions on the size of A and B under certain implicit conditions concerning the generating functions of these sets. This generalizes the earlier results of Nazarov-Perez-Volberg, Łaba-Zhai, and Bond-Volberg.

Cited by

• Sharp Favard length of random Cantor sets
• Lower bounds for mask polynomials with many cyclotomic divisors
• Favard length and generalized projections

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