Hmmmm, apparently a little bit is already known about Buffon’s noodle, after all. I will have to look at this more closely.

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Buffon’s Noodle

April 7, 2009

I almost forgot to mention the paper that myself and my advisor, Alexander Volberg, have out. Here it is.

In my post about Kakeya sets in R^2, I talked about the Buffon needle probability of a certain square Cantor set. The square Cantor set you end up with at the “end” of the construction is an abstraction, and not something one can draw a complete picture of by hand, as is the case with its corresponding Kakeya set. However, by a fact called “continuity of measures”, we know that partial constructions get successively closer to having zero needle probability/area (in the Cantor/Kakeya side of the picture, respectively), in fact converging to zero in the limit as n goes to infinity. One may then ask how quickly this convergence happens, so that, for example, one may actually construct by hand an actual set with footlong line segments in every direction whose total area is, say, less than .002 square feet.

Let S_n be the square Cantor set, and Buf(S_n) be its needle probability (in papers, this is also called the Favard length at times). It is known that there are two constants c and C such that, for example, clogn/n < Buf(S_n) < C/(n^(1/7)).

(Each inequality is a separate paper; clogn/n paper is a lot easier).

Ok. Now here’s where I come in. It seems as if there should be little difference between Buffon’s needle toss and Buffon’s ring toss when the radius of the ring is large. For the purposes of the estimate given by Bateman and Volberg, there is no difference at all when the radius is large enough compared to the fineness of the set in question. Additionally, we can generalize the question further: non-circular rigid curves (“noodles”) may be tossed at the set, landing in random places and with random orientations. Then the “noodle probability” Bufnood(S_n) > clogn/n, as long as the noodles aren’t too curvy, which we quantify in the paper. The result, when applied to circular arcs, says that a radius of r>4^(n/5) is large enough for the Bateman-Volberg estimate Bufnood(S_n) > clogn/n to hold.

An improved version of our paper is in the works which says nothing more about general noodles, but contains simpler special arguments which vastly improve the situation for circles: now r=Cn is known to be sufficient. The newer paper will also be simplified, reorganized, and have more pictures. It will also briefly talk about a certain conjecture about a modified Buffon’s ring toss which would imply the sharpness of the LloglogL boundedness of the lacunary circular maximal operator. I’ll let you know if it’s ever available online.

I hope this rant was Level 2 except for the maximal operator stuff at the end. The papers linked are a moderately light Level 3, and another level 3 that has lost me a fair amount of sleep.

Also, nothing much seems to be known about upper bounds for buffon noodle probability. I am not optimistic that it will be easy to modify existing arguments, but perhaps we will try sometime.