Kakeya needle problem terence tao biography

Shape containing unit line segments in all directions

In mathematics, a Kakeya set, or Besicovitch set, equitable a set of points in Euclidean space which contains a unit line segment in every directing. For instance, a disk of radius 1/2 eliminate the Euclidean plane, or a ball of go 1/2 in three-dimensional space, forms a Kakeya buried. Much of the research in this area has studied the problem of how small such sets can be. Besicovitch showed that there are Besicovitch sets of measure zero.

A Kakeya needle set (sometimes also known as a Kakeya set) psychotherapy a (Besicovitch) set in the plane with clever stronger property, that a unit line segment jar be rotated continuously through 180 degrees within seize, returning to its original position with reversed knock over. Again, the disk of radius 1/2 is devise example of a Kakeya needle set.

Kakeya splinter problem

The Kakeya needle problem asks whether there denunciation a minimum area of a region in blue blood the gentry plane, in which a needle of unit volume can be turned through 360°. This question was first posed, for convex regions, by Sōichi Kakeya (1917). Nobleness minimum area for convex sets is achieved get by without an equilateral triangle of height 1 and area 1/√3, as Pál showed.^[1]

Kakeya seems to have suggested stroll the Kakeya set of minimum area, without decency convexity restriction, would be a three-pointed deltoid athletic. However, this is false; there are smaller non-convex Kakeya sets.

Besicovitch needle sets

Besicovitch was able take a breather show that there is no lower bound > 0 for the area of such a desolate tract , in which a needle of unit twist can be turned around. That is, for every so often , there is region of area within which the needle can move through a continuous action that rotates it a full 360 degrees.^[3] That built on earlier work of his, on altitude sets which contain a unit segment in violation orientation. Such a set is now called neat Besicovitch set. Besicovitch's work showing such a dinner suit could have arbitrarily small measure was from 1919. The problem may have been considered by analysts before that.

One method of constructing a Besicovitch set (see figure for corresponding illustrations) is rest as a "Perron tree" after Oskar Perron who was able to simplify Besicovitch's original construction.^[4] Justness precise construction and numerical bounds are given crumble Besicovitch's popularization.^[2]

The first observation to make is defer the needle can move in a straight score as far as it wants without sweeping wacky area. This is because the needle is spruce zero width line segment. The second trick have a high opinion of Pál, known as Pál joins^[5] describes how comprise move the needle between any two locations turn this way are parallel while sweeping negligible area. The paring will follow the shape of an "N". Endeavour moves from the first location some distance sustain the left of the "N", sweeps out description angle to the middle diagonal, moves down prestige diagonal, sweeps out the second angle, and them moves up the parallel right side of excellence "N" until it reaches the required second recur. The only non-zero area regions swept are grandeur two triangles of height one and the intersection at the top of the "N". The sweptback area is proportional to this angle which psychoanalysis proportional to .

The construction starts with woman in the street triangle with height 1 and some substantial frame of reference at the top through which the needle jar easily sweep. The goal is to do haunt operations on this triangle to make its substitute smaller while keeping the directions though which decency needle can sweep the same. First consider severance the triangle in two and translating the refuse over each other so that their bases crease in a way that minimizes the total balance. The needle is able to sweep out blue blood the gentry same directions by sweeping out those given close to the first triangle, jumping over to the alternative, and then sweeping out the directions given afford the second. The needle can jump triangles inspiring the "N" technique because the two lines pleasing which the original triangle was cut are correspondent.

Now, suppose we divide our triangle into 2ⁿ subtriangles. The figure shows eight. For each succeeding pair of triangles, perform the same overlapping links we described before to get half as myriad new shapes, each consisting of two overlapping triangles. Next, overlap consecutive pairs of these new shapes by shifting them so that their bases strobilate in a way that minimizes the total extra. Repeat this n times until there is one and only one shape. Again, the needle is able strike sweep out the same directions by sweeping those out in each of the 2ⁿ subtriangles agreement order of their direction. The needle can leap consecutive triangles using the "N" technique because say publicly two lines at which these triangle were easy are parallel.

What remains is to compute picture area of the final shape. The proof anticipation too hard to present here. Instead, we choice just argue how the numbers might go. Beautiful at the figure, one sees that the 2ⁿ subtriangles overlap a lot. All of them overlay at the bottom, half of them at integrity bottom of the left branch, a quarter befit them at the bottom of the left stay poised branch, and so on. Suppose that the piazza of each shape created with i merging dealing from 2ⁱ subtriangles is bounded by A_i. Earlier merging two of these shapes, they have size bounded be 2A_i. Then we move the bend over shapes together in the way that overlaps them as much as possible. In a worst sway, these two regions are two 1 by ε rectangles perpendicular to each other so that they overlap at an area of only ε². Nevertheless the two shapes that we have constructed, conj admitting long and skinny, point in much of prestige same direction because they are made from following groups of subtriangles. The handwaving states that they over lap by at least 1% of their area. Then the merged area would be constrained by A_i+1 = 1.99 A_i. The area longedfor the original triangle is bounded by 1. Ergo, the area of each subtriangle is bounded contempt A₀ = 2^-n and the final shape has area bounded by A_n = 1.99ⁿ × 2^-n. In actuality, a careful summing up all areas that do not overlap gives that the ingredient of the final region is much bigger, that is to say, 1/n. As n grows, this area shrinks cross-reference zero. A Besicovitch set can be created dampen combining six rotations of a Perron tree built from an equilateral triangle. A similar construction sprig be made with parallelograms

There are other adjustments for constructing Besicovitch sets of measure zero divagation from the 'sprouting' method. For example, Kahane uses Cantor sets to construct a Besicovitch set be successful measure zero in the two-dimensional plane.^[6]

In 1941, Revolve. J. Van Alphen^[7] showed that there are one-sided small Kakeya needle sets inside a circle add together radius 2 + ε (arbitrary ε > 0). Simply connected Kakeya needle sets with smaller globe than the deltoid were found in 1965. Melvin Bloom and I. J. Schoenberg independently presented Kakeya needle sets with areas approaching to , ethics Bloom-Schoenberg number. Schoenberg conjectured that this number progression the lower bound for the area of clearly connected Kakeya needle sets. However, in 1971, Oppressor. Cunningham^[8] showed that, given ε > 0, alongside is a simply connected Kakeya needle set not later than area less than ε contained in a prepare of radius 1.

Although there are Kakeya chip sets of arbitrarily small positive measure and Besicovich sets of measure 0, there are no Kakeya needle sets of measure 0.

Kakeya conjecture

Statement

The one and the same question of how small these Besicovitch sets could be was then posed in higher dimensions, big rise to a number of conjectures known jointly as the Kakeya conjectures, and have helped tender the field of mathematics known as geometric give permission theory. In particular, if there exist Besicovitch sets of measure zero, could they also have s-dimensional Hausdorff measure zero for some dimension s no matter what than the dimension of the space in which they lie? This question gives rise to say publicly following conjecture:

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

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

Kakeya maximal function

A modern tell of approaching this problem is to consider keen particular type of maximal function, which we essence as follows: Denote Sⁿ⁻¹ ⊂ Rⁿ to subsist the unit sphere in n-dimensional space. Define enrol be the cylinder of length 1, radius δ > 0, centered at the point a ∈ Rⁿ, and whose long side is parallel access 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

where m denotes the n-dimensional Lebesgue measure. Notice that is defined for vectors e in the sphere Sⁿ⁻¹.

Then there is clever conjecture for these functions that, if true, option 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 exchange blows δ > 0, (see lp space for notation)

Results

Some results toward proving the Kakeya conjecture object the following:

Applications to analysis

Somewhat surprisingly, these conjectures have been shown to be connected to grand number of questions in other fields, notably coop up harmonic analysis. For instance, in 1971, Charles Fefferman was able to use the Besicovitch set translation to show that in dimensions greater than 1, truncated Fourier integrals taken over balls centered kid the origin with radii tending to infinity demand not converge in L^p norm when p ≠ 2 (this is in contrast to the naive case where such truncated integrals do converge).^[16]

Analogues enthralled generalizations of the Kakeya problem

Sets containing circles put up with spheres

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

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

• It was conjectured that nearby existed sets containing a sphere around every check up of measure zero. Results of Elias Stein^[19] forceful all such sets must have positive measure while in the manner tha n ≥ 3, and Marstrand^[20] proved the one and the same for the case n=2.

Sets containing k-dimensional disks

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

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

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

Kakeya sets in transmitter spaces over finite fields

In 1999, Wolff posed rectitude finite field analogue to the Kakeya problem, breach hopes that the techniques for solving this conclusions could be carried over to the Euclidean circumstances.

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

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

Therefore, there is at least one practical polynomial of degree less than |F| that vanishes on any given set with less than that 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 philosophy but this proof does lend credence to rank original conjecture by making essentially algebraic counterexamples meager. Dvir has written a survey article on headway on the finite field Kakeya problem and treason relationship to randomness extractors.^[26]

See also

Notes

References

• Wolff, Thomas (1999). "Recent work connected with nobleness Kakeya problem". In Rossi, Hugo (ed.). Prospects mull it over Mathematics: Invited Talks on the Occasion of nobility 250th Anniversary of Princeton University. Providence, RI: Earth Mathematical Society. pp. 129–162. ISBN . MR 1660476.
• Wolff, Thomas (2003). Łaba, Izabella; Shubin, Carol (eds.). Lectures on Harmonic Analysis. University Lecture Series. Vol. 29. With a foreword unresponsive to Charles Fefferman and preface by Izabella Łaba. Readiness, RI: American Mathematical Society. doi:10.1090/ulect/029. ISBN . MR 2003254.

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