What is your work?

November 22, 2010

The Killam grant at the University of British Columbia is a general postdoctoral grant offered to students of all areas of study. As such, it is evaluated by non-mathematicians. So the application materials are meant to be read by a more general audience. So here is one of the grant proposal files, describing what I hope to be working on next year.

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Our new, improved paper is online now. Not only is the estimate much sharper thanks to one nice little lemma using Hardy space theory and another using something we call “analytic tiling”*, but you may even be able to read this one.

A 3^{-n}-neighborhood of a 1-dimensional Sierpinski gasket decays in Buffon needle probability at least as fast as C/{n^p}.

* – “Analytic tiling” is essentially our fancy name for the observation that if three unit complex numbers sum to 0, they form an equilateral triangle, and therefore, their third powers sum to a number of size 3. The name is motivated by the role tiling played in Laba and Zhai’s paper on product Cantor sets.

The 3^{-n}-neighborhood of the 1-dimensional Sierpinski’s gasket decays in Buffon needle probability at least as fast C/n^p, for some p>0.

Alexander Volberg will probably be posting our result soon. It’s like the last one, only better.

The complex analysis involved is relatively elementary, but intricate. But at any rate, Dr. Alexander Volberg and I now have an analog of the Nazarov, Peres, Volberg result, adapted to the Sierpinski gasket with a moderate loss.

Here it is. I have it labelled as a Level 3, but this one will probably keep you up late regardless of who you are or how much you know.

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

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.

Kakeya Sets in R^2

March 23, 2009

Kakeya sets: Unit Length Line Segments In Every Direction In The x-y Plane Cleverly Arranged To Take Up Zero Area

Abstract. A Kakeya set in the x-y plane is a set containing unit length line segments in all possible directions cleverly arranged to overlap so much that together they take up zero area. If we think of the slopes and y-intercepts as ordered pairs and make a graph of them, one such possible graph is given by a rotated copy of a 2-dimensional Cantor set. The fact that this Kakeya set has zero area corresponds exactly to the fact that this Cantor set has zero Bu ffon needle probability, which in turn is a consequence of a deep theorem of Abram Besicovitch. 1-dimensional Hausdorff measure is briefly discussed.

(Click here for the most recent version of the paper)

(Older draft, v. 3-2)

(Older draft, v. 3-1)

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I recently wrote this paper for my own benefit while preparing a talk for a student conference. It is a difficult task to condense to a 20-minute talk something that is demonstrably a 9-page undertaking.

The target audience of the talk (and hence the paper) is a mixed audience of undergraduates and grad students. Ideally, it should be understandable by a motivated High School student who is competent in plane geometry and has a pretty good idea of what an integral is. Most of the paper has been generalized elsewhere, but this has been intentionally left out for simplicity.

The paper may be updated later. I was thinking about adding a few exercises to the paper for fun. For example, look at equation (2.2). It maybe be difficult to apply this formula for general sets S, but it’s manageable for a couple of cases: S is a line segment, and S is a circle. Of course, one should figure out what set K in the plane one gets from the line segment set and find its area by more conventional means for comparison.

And to think that when I took Calc 1, I had just assumed that the integral of csc-cubed would never come up naturally!

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EDIT: There are slight variations on this problem. For example, one can ask what the situation is if one is required to be able to continuously maneuver a single line segment to take on all orientations as it moves within the planar set K, and in fact, this is closer to the the problem as originally posed by Kakeya. Terry Tao addresses this analog using only the usual tools and language of analysis that should make sense to any fluent veteran of Walter Rudin’s Principles of Mathematical Analysis.

EDIT 2: I presented the content of this paper at MSU’s 2009 Student Research Conference a couple weeks ago, and won third prize.