Some very small edits have been made to the paper on Kakeya sets in R^2 – typos fixed, small clarifications made, small errors corrected. Some distracting comments were made into footnotes.

Future editions of this primer might include another drawing or two, suggested exercises, and a remark or two about how the well-known content of this paper relates to my current research.

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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.