The Kakeya problem. The University of Manchester. Jonathan Fraser
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1 Jonathan M. Fraser The University of Manchester
2 Kakeya needle sets A subset of the plane is called a Kakeya needle set if a unit line segment can be smoothly rotated within it by 360 degrees.
3 Kakeya needle sets A subset of the plane is called a Kakeya needle set if a unit line segment can be smoothly rotated within it by 360 degrees. For example, the closed ball B(0, 1/2) is a Kakeya needle set
4 Kakeya needle sets A subset of the plane is called a Kakeya needle set if a unit line segment can be smoothly rotated within it by 360 degrees. For example, the closed ball B(0, 1/2) is a Kakeya needle set
5 Kakeya needle sets A subset of the plane is called a Kakeya needle set if a unit line segment can be smoothly rotated within it by 360 degrees. For example, the closed ball B(0, 1/2) is a Kakeya needle set Question: How small (in area) can Kakeya needle sets be?
6 Kakeya needle sets A subset of the plane is called a Kakeya needle set if a unit line segment can be smoothly rotated within it by 360 degrees. For example, the closed ball B(0, 1/2) is a Kakeya needle set Question: How small (in area) can Kakeya needle sets be? Or, what is the minimal comfortable living conditions for a unit worm?
7 Kakeya needle sets A subset of the plane is called a Kakeya needle set if a unit line segment can be smoothly rotated within it by 360 degrees. For example, the closed ball B(0, 1/2) is a Kakeya needle set Question: How small (in area) can Kakeya needle sets be? Or, what is the minimal comfortable living conditions for a unit worm?
8 Kakeya needle sets Clearly, a Kakeya needle set must have positive area...
9 Kakeya needle sets Clearly, a Kakeya needle set must have positive area... But Besicovitch proved in 1919 that one can find examples with arbitrarily small area!
10 Kakeya needle sets
11 Kakeya sets A subset K of R d is called a Kakeya set if it contains a unit line segment in every direction.
12 Kakeya sets A subset K of R d is called a Kakeya set if it contains a unit line segment in every direction. Question: Can Kakeya sets have zero volume?
13 Kakeya sets A subset K of R d is called a Kakeya set if it contains a unit line segment in every direction. Question: Can Kakeya sets have zero volume? Answer: Yes! Besicovitch modified his previous result to prove this in 1928.
14 Kakeya sets A subset K of R d is called a Kakeya set if it contains a unit line segment in every direction. Question: Can Kakeya sets have zero volume? Answer: Yes! Besicovitch modified his previous result to prove this in Question: Can they be even smaller?
15 Kakeya sets A subset K of R d is called a Kakeya set if it contains a unit line segment in every direction. Question: Can Kakeya sets have zero volume? Answer: Yes! Besicovitch modified his previous result to prove this in Question: Can they be even smaller? What does this even mean?
16 Dimension and fractals A fractal is a highly irregular and complicated set with detail on all scales
17 Dimension and fractals A fractal is a highly irregular and complicated set with detail on all scales - Kakeya sets can be fractal!
18 Dimension and fractals A fractal is a highly irregular and complicated set with detail on all scales - Kakeya sets can be fractal! Classic notions of dimension and measure do not apply to fractals and so we have to invent new ones!
19 Dimension and fractals Given a length scale δ > 0, let N δ (F ) be the minimum number of sets of diameter δ required to cover a set F.
20 Dimension and fractals Given a length scale δ > 0, let N δ (F ) be the minimum number of sets of diameter δ required to cover a set F. For simple geometric objects we see that N δ (F ) δ dimension
21 Dimension and fractals Given a length scale δ > 0, let N δ (F ) be the minimum number of sets of diameter δ required to cover a set F. For simple geometric objects we see that N δ (F ) δ dimension This motivates, the box dimension: dim B F = lim δ 0 log N δ (F ) log δ
22 Dimension and fractals Given a length scale δ > 0, let N δ (F ) be the minimum number of sets of diameter δ required to cover a set F. For simple geometric objects we see that N δ (F ) δ dimension This motivates, the box dimension: assuming this limit exists! dim B F = lim δ 0 log N δ (F ) log δ
23 Dimension and fractals Given a length scale δ > 0, let N δ (F ) be the minimum number of sets of diameter δ required to cover a set F. For simple geometric objects we see that N δ (F ) δ dimension This motivates, the box dimension: dim B F = lim δ 0 log N δ (F ) log δ assuming this limit exists! Otherwise we define upper and lower box dimension dim UB F and dim LB F.
24 If a set F in R d has positive volume, then it has full box dimension dim B F = d.
25 If a set F in R d has positive volume, then it has full box dimension dim B F = d. This motivates Question: Do all Kakeya sets have full dimension?
26 If a set F in R d has positive volume, then it has full box dimension dim B F = d. This motivates Question: Do all Kakeya sets have full dimension? The real Kakeya problem uses Hausdorff dimension, but we will use box dimension today.
27 If a set F in R d has positive volume, then it has full box dimension dim B F = d. This motivates Question: Do all Kakeya sets have full dimension? The real Kakeya problem uses Hausdorff dimension, but we will use box dimension today. This is often referred to as the weak Kakeya problem.
28 What is known?
29 What is known? 1971: Roy Davies proved that all Kakeya sets in the plane have Hausdorff dimension 2.
30 What is known? 1971: Roy Davies proved that all Kakeya sets in the plane have Hausdorff dimension : Wolff proved that all Kakeya sets in R d have Hausdorff dimension at least d/2 + 1.
31 What is known? 1971: Roy Davies proved that all Kakeya sets in the plane have Hausdorff dimension : Wolff proved that all Kakeya sets in R d have Hausdorff dimension at least d/ : Katz and Tao proved that all Kakeya sets in R d have Hausdorff dimension at least (2 2)d + (4 2 5) 0.585d
32 What is known? 1971: Roy Davies proved that all Kakeya sets in the plane have Hausdorff dimension : Wolff proved that all Kakeya sets in R d have Hausdorff dimension at least d/ : Katz and Tao proved that all Kakeya sets in R d have Hausdorff dimension at least (2 2)d + (4 2 5) 0.585d : Katz, Tao and Laba proved that all Kakeya sets in R 3 have box dimension at least 5/2 + ε
33 What is known? 1971: Roy Davies proved that all Kakeya sets in the plane have Hausdorff dimension : Wolff proved that all Kakeya sets in R d have Hausdorff dimension at least d/ : Katz and Tao proved that all Kakeya sets in R d have Hausdorff dimension at least (2 2)d + (4 2 5) 0.585d : Katz, Tao and Laba proved that all Kakeya sets in R 3 have box dimension at least 5/2 + ε = 3/ ε.
34 My co-authors
35
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