The Favard length of product Cantor sets

Abstract

Nazarov, Peres and Volberg proved recently in [8] that the Favard length of the n-th iteration of the four-corner Cantor set is bounded from above by n −c for an appropriate c. We generalize this result to all product Cantor sets whose projection in some direction has positive 1-dimensional measure.

Buffon’s needle estimates for rational product Cantor sets

“…A does not contribute to the small values of P 2 and can be safely ignored. The factor φ (1) A has the SSV property; this was used in a weaker and somewhat camouflaged form in [12], [7], [3]. Furthermore, we will prove in Proposition 4.3 that φ…”

“…The proof of Theorem 1.2 is based on a new method of estimating socalled "Riesz products" of trigonometric polynomials. The arguments of [12], with the additional modifications of [3], [4], [7], have reduced the problem to proving lower bounds on integrals of the form For general (not necessarily product) self-similar sets, we defiine φ θ (ξ) = 1 L L j=1 e 2πir j cos(θ j −θ) instead, where z j = r j e 2πiθ j are the similarity centers. The strategy of [12], [3], [4], [7] is now as follows.…”

Recent Progress on Favard Length Estimates for Planar Cantor Sets

“…• the rational product sets in Example 3 under the additional "tiling" condition that |proj θ 0 (E ∞ )| > 0 for some direction θ 0 ( Laba-Zhai [11]).…”

“…In Section 3, we begin to work towards proving the main estimate. The reductions in Section 3.1 and Lemma 3.3 are due to [18], with minor modifications in [4], [5], [11]. The remaining issue concerns integrating a certain exponential sum on a set where another exponential sum, which we will call |P 2 (ξ)| 2 , is known to be bounded from below away from 0.…”

Buffon needle lands in ϵ-neighborhood of a 1-dimensional Sierpinski Gasket with probability at most |logϵ|−c

“…Recently one detects a considerable interest in estimating the Favard length of such -neighborhoods of Besicovitch irregular sets, see [5,6,4,3]. In [5] a random model of such Cantor set is considered and the estimate 1 n infinitely often, almost surely is proved.…”