Kakeya set

The same question of how small these Besicovitch sets could be was then posed in higher dimensions, giving rise to a number of conjectures known collectively as the Kakeya conjectures, and have helped initiate the field of mathematics known as geometric measure theory. In particular, if there exist Besicovitch sets of measure zero, could they also have s-dimensional Hausdorff measure zero for some dimension s less than the dimension of the space in which they lie? This question gives rise to the following conjecture:

Kakeya set conjecture: Define a Besicovitch set in Rⁿ to be a set which contains a unit line segment in every direction. Is it true that such sets necessarily have Hausdorff dimension and Minkowski dimension equal to n?

This is known to be true for n = 1, 2 but only partial results are known in higher dimensions.

A modern way of approaching this problem is to consider a particular type of maximal function, which we construct as follows: Denote Sⁿ⁻¹ ⊂ Rⁿ to be the unit sphere in n-dimensional space. Define ${\displaystyle T_{e}^{\delta }(a)}$ to be the cylinder of length 1, radius δ > 0, centered at the point a ∈ Rⁿ, and whose long side is parallel to the direction of the unit vector e ∈ Sⁿ⁻¹. Then for a locally integrable function f, we define the Kakeya maximal function of f to be

${\displaystyle f_{*}^{\delta }(e)=\sup _{a\in \mathbf {R} ^{n}}{\frac {1}{m(T_{e}^{\delta }(a))}}\int _{T_{e}^{\delta }(a)}|f(y)|dm(y)}$

where m denotes the n-dimensional Lebesgue measure. Notice that ${\displaystyle f_{*}^{\delta }}$ is defined for vectors e in the sphere Sⁿ⁻¹.

Then there is a conjecture for these functions that, if true, will imply the Kakeya set conjecture for higher dimensions:

Kakeya maximal function conjecture: For all ε > 0, there exists a constant C_ε > 0 such that for any function f and all δ > 0, (see lp space for notation)

${\displaystyle \left\|f_{*}^{\delta }\right\|_{L^{n}(\mathbf {S} ^{n-1})}\leqslant C_{\epsilon }\delta ^{-\epsilon }\|f\|_{L^{n}(\mathbf {R} ^{n})}.}$

Some results toward proving the Kakeya conjecture are the following:

• The Kakeya conjecture is true for n = 1 (trivially) and n = 2 (Davies[9]).
• In any n-dimensional space, Wolff[10] showed that the dimension of a Kakeya set must be at least (n+2)/2.
• In 2002, Katz and Tao[11] improved Wolff's bound to ${\displaystyle (2-{\sqrt {2}})(n-4)+3}$, which is better for n > 4.
• In 2000, Katz, Łaba, and Tao[12] proved that the Minkowski dimension of Kakeya sets in 3 dimensions is strictly greater than 5/2.
• In 2000, Jean Bourgain connected the Kakeya problem to arithmetic combinatorics[13][14] which involves harmonic analysis and additive number theory.
• In 2017, Katz and Zahl[15] improved the lower bound on the Hausdorff dimension of Besicovitch sets in 3 dimensions to ${\displaystyle 5/2+\epsilon }$ for an absolute constant ${\displaystyle \epsilon >0}$.

Analogues of the Kakeya problem include considering sets containing more general shapes than lines, such as circles.

• In 1997[17] and 1999,[18] Wolff proved that sets containing a sphere of every radius must have full dimension, that is, the dimension is equal to the dimension of the space it is lying in, and proved this by proving bounds on a circular maximal function analogous to the Kakeya maximal function.

• It was conjectured that there existed sets containing a sphere around every point of measure zero. Results of Elias Stein[19] proved all such sets must have positive measure when n ≥ 3, and Marstrand[20] proved the same for the case n=2.

A generalization of the Kakeya conjecture is to consider sets that contain, instead of segments of lines in every direction, but, say, portions of k-dimensional subspaces. Define an (n, k)-Besicovitch set K to be a compact set in Rⁿ containing a translate of every k-dimensional unit disk which has Lebesgue measure zero. That is, if B denotes the unit ball centered at zero, for every k-dimensional subspace P, there exists x ∈ Rⁿ such that (P ∩ B) + x ⊆ K. Hence, a (n, 1)-Besicovitch set is the standard Besicovitch set described earlier.

The (n, k)-Besicovitch conjecture: There are no (n, k)-Besicovitch sets for k > 1.

In 1979, Marstrand[21] proved that there were no (3, 2)-Besicovitch sets. At around the same time, however, Falconer[22] proved that there were no (n, k)-Besicovitch sets for 2k > n. The best bound to date is by Bourgain,[23] who proved in that no such sets exist when 2^k−1 + k > n.

In 1999, Wolff posed the finite field analogue to the Kakeya problem, in hopes that the techniques for solving this conjecture could be carried over to the Euclidean case.

Finite Field Kakeya Conjecture: Let F be a finite field, let K ⊆ Fⁿ be a Kakeya set, i.e. for each vector y ∈ Fⁿ there exists x ∈ Fⁿ such that K contains a line {x + ty : t ∈ F}. Then the set K has size at least c_n|F|ⁿ where c_n>0 is a constant that only depends on n.

Zeev Dvir proved this conjecture in 2008, showing that the statement holds for c_n = 1/n!.[24][25] In his proof, he observed that any polynomial in n variables of degree less than |F| vanishing on a Kakeya set must be identically zero. On the other hand, the polynomials in n variables of degree less than |F| form a vector space of dimension

${\displaystyle {|\mathbf {F} |+n-1 \choose n}\geq {\frac {|\mathbf {F} |^{n}}{n!}}.}$

Therefore, there is at least one non-trivial polynomial of degree less than |F| that vanishes on any given set with less than this number of points. Combining these two observations shows that Kakeya sets must have at least |F|ⁿ/n! points.

It is not clear whether the techniques will extend to proving the original Kakeya conjecture but this proof does lend credence to the original conjecture by making essentially algebraic counterexamples unlikely. Dvir has written a survey article on recent progress on the finite field Kakeya problem and its relationship to randomness extractors.[26]