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



April 12, 2009

Interesting. WordPress logs a lot of stats, and tracks where people click from to get to your wordpress blog. In particular, I linked to an entry on Terry Tao’s blog as an example of what a “level 1” math post might look like, and that day I got 40-something visits from his blog, followed by the usual near-complete obscurity. It seems that whenever someone links to a wordpress blog, it manufactures a comment on the target’s blog. This is interesting, though kind of shady. At any rate, I’ll try to use this knowledge sparingly but intentionally, hopefully striking a balance between relevance and self-promotion. If you follow a link over here from such a place, let me know if you like what you see. Comments are actually quite welcome.

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.

Prerequisite levels

April 7, 2009

I’ve decided to set up a prerequisite ranking scheme for the content of this blog, so as to possibly reach a broader audience and keep the scary stuff tucked away where it can be read later or not at all. My current plan is to group them into 4 levels, with some inevitable overlap. Those of you who self-diagnose as belonging at level 2-4 shouldn’t necessarily expect an easy read, except for at levels strictly below your level.

Level 1 Content – A post with this label is intended to be for a general audience.  It might look something like this. I don’t plan on writing a lot of these, but we will see.

Level 2 Content – A post with this label is intended for a somewhat technical audience. Since what I study is analysis, I will probably have a hard time finding anything to talk about that doesn’t at least require the reader to know what an integral and a derivative are. One or two semesters of Calculus in college should be enough, if you feel like you really got it. I might double-state things in formal and informal language for the benefit of those not belonging to higher levels, even if it makes things a little wordy.

Level 3 Content – A post with this label is at the level of a general graduate school math course. The reader should have knowledge of some typical courses a graduate student has to take to pass qualifying exams. Most likely real and complex analysis, but if other stuff plays a big role, I’ll be sure to give fair warning.

Level 4 Content – I am not yet qualified to write posts at this level, though I’m trying to remedy this in the moderately near future.

Here’s something that should be in every complex analysis text, but aparently isn’t. But it’s here on my blog, so never fear!

You know the boundary values of a holomorphic function f on the disc, and you know one of its values on the interior. How many zeroes can there be? And how tightly can you quarantine away the set where |f| is small? Blaschke products, Harnack’s inequality, and the maximum modulus principle used together provide one partial answer.

Click here for the short paper.