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Article overview
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Buffon's needle landing near Besicovitch irregular self-similar sets | | Matt Bond
; Alexander Volberg
; | | Date: |
27 Dec 2009 | | Abstract: | In this paper we get an estimate of Favard length of an arbitrary
neighbourhood of an arbitrary self-similar Cantor set. Consider $L$ closed
disjoint discs of radius $1/L$ inside the unit disc. By using linear maps of
smaller disc onto the unit disc we can generate a self-similar Cantor set $G$.
Then $G=igcap_nG_n$. One may then ask the rate at which the Favard length
-- the average over all directions of the length of the orthogonal projection
onto a line in that direction -- of these sets $G_n$ decays to zero as a
function of $n$. The quantitative results for the Favard length problem were
obtained by Peres--Solomyak and Tao; in the latter paper a general way of
making a quantitative statement from the Besicovitch theorem is considered. But
being rather general, this method does not give a good estimate for
self-similar structures such as $G_n$. Indeed, vastly improved estimates have
been proven in these cases: in the paper of Nazarov--Peres--Volberg, it was
shown that for 1/4 corner Cantor set one has $p<1/6$, such that
$Fav(K_n)leqfrac{c_p}{n^{p}}$, and in Laba--Zhai and Bond--Volberg the same
type power estimate was proved for the product Cantor sets (with an extra
tiling property) and for the Sierpinski gasket $S_n$ for some other $p>0$. In
the present work we give an estimate that works for {it any} Besicovitch set
which is self-similar. However estimate is worse than the power one. The power
estimate still appears to be related to a certain regularity property of zeros
of a corresponding linear combination of exponents (we call this property {it
analytic tiling}). | | Source: | arXiv, 0912.5111 | | Services: | Forum | Review | PDF | Favorites | |
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