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One of my favourite unsolved problems in mathematics is the Kakeya conjecture in geometric measure theory. This conjecture is descended from the
Kakeya needle problem. (1917) What is the least area in the plane required to continuously rotate a needle of unit length and zero thickness around completely (i.e. by )?
For instance, one can rotate a unit needle inside a unit disk, which has area . By using a deltoid one requires only area.
In 1928, Besicovitch showed that that in fact one could rotate a unit needle using an arbitrarily small amount of positive area. This unintuitive fact was a corollary of two observations. The first, which is easy, is that one can translate a needle using arbitrarily small area, by sliding the needle along the direction it points in for a long distance (which costs zero area), turning it slightly (costing a small amount of area), sliding back, and then undoing the turn. The second fact, which is less obvious, can be phrased as follows. Define a Kakeya set in to be any set which contains a unit line segment in each direction. (See this Java applet of mine, or the wikipedia page, for some pictures of such sets.)
Theorem. (Besicovitch, 1919) There exists Kakeya sets of arbitrarily small area (or more precisely, Lebesgue measure).
In fact, one can construct such sets with zero Lebesgue measure. On the other hand, it was shown by Davies that even though these sets had zero area, they were still necessarily two-dimensional (in the sense of either Hausdorff dimension or Minkowski dimension). This led to an analogous conjecture in higher dimensions:
Kakeya conjecture. A Besicovitch set in (i.e. a subset of that contains a unit line segment in every direction) has Minkowski and Hausdorff dimension equal to n.
This conjecture remains open in dimensions three and higher (and gets more difficult as the dimension increases), although many partial results are known. For instance, when n=3, it is known that Besicovitch sets have Hausdorff dimension at least 5/2 and (upper) Minkowski dimension at least . See my Notices article for a general survey of this problem (and its connections with Fourier analysis, additive combinatorics, and PDE), my paper with Katz for a more technical survey, and Wolff’s survey for a systematic treatment of the field (up to about 1998 or so).
In 1999, Wolff proposed a simpler finite field analogue of the Kakeya conjecture as a model problem that avoided all the technical issues involving Minkowski and Hausdorff dimension. If is a vector space over a finite field F, define a Kakeya set to be a subset of which contains a line in every direction.
Finite field Kakeya conjecture. Let be a Kakeya set. Then E has cardinality at least , where depends only on n.
This conjecture has had a significant influence in the subject, in particular inspiring work on the sum-product phenomenon in finite fields, which has since proven to have many applications in number theory and computer science. Modulo minor technicalities, the progress on the finite field Kakeya conjecture was, until very recently, essentially the same as that of the original “Euclidean” Kakeya conjecture.
Last week, the finite field Kakeya conjecture was proven using a beautifully simple argument by Zeev Dvir, using the polynomial method in algebraic extremal combinatorics. The proof is so short that I can present it in full here.