[PPT] - New constructions of Kakeya and Besicovitch sets Yuval Peres 1 Based PowerPoint Presentation

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New constructions of Kakeya and Besicovitch sets

Yuval Peres 1

Based on work with

Y. Babichenko, R. Peretz, P. Sousi, P. Winkler and

the forthcoming book Fractals in Probability and Analysis with C. Bishop

1Microsoft Research

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Besicovitch sets – History

A subset S ⊆ R2 is called a Besicovitch set if it contains a unit segment in every direction. It is called a Kakeya set if a unit segment can be rotated 360 degrees within S.

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Besicovitch sets – History

A subset S ⊆ R2 is called a Besicovitch set if it contains a unit segment in every direction. It is called a Kakeya set if a unit segment can be rotated 360 degrees within S. Kakeya’s question (1917): Does the three-pointed deltoid shape have minimal area among such sets?

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Besicovitch and Schoenberg’s constructions

Besicovitch (1919) gave the first construction of a Besicovitch set of zero area.

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Besicovitch and Schoenberg’s constructions

Besicovitch (1919) gave the first construction of a Besicovitch set of zero area. Due to a reduction by P´ al, this also yields a Kakeya set of arbitrarily small area.

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Besicovitch and Schoenberg’s constructions

Besicovitch (1919) gave the first construction of a Besicovitch set of zero area. Due to a reduction by P´ al, this also yields a Kakeya set of arbitrarily small area. Besicovitch’s construction was later simplified by Perron and Schoenberg who gave a construction of a Besicovitch set consisting of 4n triangles of area of

rder 1/ log n.

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Besicovitch and Schoenberg’s constructions

Besicovitch (1919) gave the first construction of a Besicovitch set of zero area. Due to a reduction by P´ al, this also yields a Kakeya set of arbitrarily small area. Besicovitch’s construction was later simplified by Perron and Schoenberg who gave a construction of a Besicovitch set consisting of 4n triangles of area of

rder 1/ log n.

n = 1

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Besicovitch and Schoenberg’s constructions

Besicovitch (1919) gave the first construction of a Besicovitch set of zero area. Due to a reduction by P´ al, this also yields a Kakeya set of arbitrarily small area. Besicovitch’s construction was later simplified by Perron and Schoenberg who gave a construction of a Besicovitch set consisting of 4n triangles of area of

rder 1/ log n.

n = 1 n = 2

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Besicovitch and Schoenberg’s constructions

Besicovitch (1919) gave the first construction of a Besicovitch set of zero area. Due to a reduction by P´ al, this also yields a Kakeya set of arbitrarily small area. Besicovitch’s construction was later simplified by Perron and Schoenberg who gave a construction of a Besicovitch set consisting of 4n triangles of area of

rder 1/ log n.

n = 1 n = 2 n = 4

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Besicovitch and Schoenberg’s constructions

Besicovitch (1919) gave the first construction of a Besicovitch set of zero area. Due to a reduction by P´ al, this also yields a Kakeya set of arbitrarily small area. Besicovitch’s construction was later simplified by Perron and Schoenberg who gave a construction of a Besicovitch set consisting of 4n triangles of area of

rder 1/ log n.

n = 1 n = 2 n = 4 n = 256

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Besicovitch and Schoenberg’s constructions

Besicovitch (1919) gave the first construction of a Besicovitch set of zero area. Due to a reduction by P´ al, this also yields a Kakeya set of arbitrarily small area. Besicovitch’s construction was later simplified by Perron and Schoenberg who gave a construction of a Besicovitch set consisting of 4n triangles of area of

rder 1/ log n.

n = 1 n = 2 n = 4 n = 256

Yuval Peres New constructions of Kakeya and Besicovitch sets

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New connection to game theory and probability

In this talk we will see a probabilistic construction of an optimal Besicovitch set consisting of triangle (and later a deterministic analog).

Yuval Peres New constructions of Kakeya and Besicovitch sets

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New connection to game theory and probability

In this talk we will see a probabilistic construction of an optimal Besicovitch set consisting of triangle (and later a deterministic analog). We do so by relating these sets to a game of pursuit on the cycle Zn introduced by Adler et al.

Yuval Peres New constructions of Kakeya and Besicovitch sets

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New connection to game theory and probability

In this talk we will see a probabilistic construction of an optimal Besicovitch set consisting of triangle (and later a deterministic analog). We do so by relating these sets to a game of pursuit on the cycle Zn introduced by Adler et al.

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Definition of the game Gn

Two players

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Definition of the game Gn

Two players

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Definition of the game Gn

Two players

Hunter

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Definition of the game Gn

Two players

Hunter

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Definition of the game Gn

Two players

Hunter Rabbit

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Definition of the game Gn

Two players

Hunter Rabbit

Where?

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Definition of the game Gn

Two players

Hunter Rabbit

Where?

On Zn

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Definition of the game When?

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Definition of the game When?

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Definition of the game When?

At night – they cannot see each other....

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Definition of the game Gn

Rules

rrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr hhhhhhhhhhhhh

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Definition of the game Gn

Rules

At time 0 both hunter and rabbit choose initial positions. rrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr hhhhhhhhhhhhh

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Definition of the game Gn

Rules

At time 0 both hunter and rabbit choose initial positions. At each subsequent step, the hunter either moves to an adjacent node or stays

put. Simultaneously, the rabbit may leap to any node in Zn.

rrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr hhhhhhhhhhhhh

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Definition of the game Gn

Rules

At time 0 both hunter and rabbit choose initial positions. At each subsequent step, the hunter either moves to an adjacent node or stays

put. Simultaneously, the rabbit may leap to any node in Zn.

When does the game end?

rrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr hhhhhhhhhhhhh

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Definition of the game Gn

Rules

At time 0 both hunter and rabbit choose initial positions. At each subsequent step, the hunter either moves to an adjacent node or stays

put. Simultaneously, the rabbit may leap to any node in Zn.

When does the game end?

At “capture time”, when the hunter and the rabbit occupy the same location in Zn at the same time. rrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr hhhhhhhhhhhhh

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 30

Definition of the game Gn

Rules

At time 0 both hunter and rabbit choose initial positions. At each subsequent step, the hunter either moves to an adjacent node or stays

put. Simultaneously, the rabbit may leap to any node in Zn.

When does the game end?

At “capture time”, when the hunter and the rabbit occupy the same location in Zn at the same time. rrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr hhhhhhhhhhhhh

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 31

Definition of the game Gn

Rules

At time 0 both hunter and rabbit choose initial positions. At each subsequent step, the hunter either moves to an adjacent node or stays

put. Simultaneously, the rabbit may leap to any node in Zn.

When does the game end?

At “capture time”, when the hunter and the rabbit occupy the same location in Zn at the same time.

Goals

rrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr hhhhhhhhhhhhh

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 32

Definition of the game Gn

Rules

At time 0 both hunter and rabbit choose initial positions. At each subsequent step, the hunter either moves to an adjacent node or stays

put. Simultaneously, the rabbit may leap to any node in Zn.

When does the game end?

At “capture time”, when the hunter and the rabbit occupy the same location in Zn at the same time.

Goals

Hunter: Minimize “capture time” rrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr hhhhhhhhhhhhh

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Definition of the game Gn

Rules

At time 0 both hunter and rabbit choose initial positions. At each subsequent step, the hunter either moves to an adjacent node or stays

put. Simultaneously, the rabbit may leap to any node in Zn.

When does the game end?

At “capture time”, when the hunter and the rabbit occupy the same location in Zn at the same time.

Goals

Hunter: Minimize “capture time” Rabbit: Maximize “capture time” rrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr hhhhhhhhhhhhh

Yuval Peres New constructions of Kakeya and Besicovitch sets

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The n-step game G ∗

n

Define a zero sum game G ∗

n with payoff 1 to the hunter if he captures the

rabbit in the first n steps, and payoff 0 otherwise.

Yuval Peres New constructions of Kakeya and Besicovitch sets

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The n-step game G ∗

n

Define a zero sum game G ∗

n with payoff 1 to the hunter if he captures the

rabbit in the first n steps, and payoff 0 otherwise. G ∗

n is finite ⇒ By the minimax theorem, ∃ optimal randomized

strategies for both players.

Yuval Peres New constructions of Kakeya and Besicovitch sets

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The n-step game G ∗

n

Define a zero sum game G ∗

n with payoff 1 to the hunter if he captures the

rabbit in the first n steps, and payoff 0 otherwise. G ∗

n is finite ⇒ By the minimax theorem, ∃ optimal randomized

strategies for both players. The value of G ∗

n is the probability pn of capture under optimal play.

Yuval Peres New constructions of Kakeya and Besicovitch sets

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The n-step game G ∗

n

Define a zero sum game G ∗

n with payoff 1 to the hunter if he captures the

rabbit in the first n steps, and payoff 0 otherwise. G ∗

n is finite ⇒ By the minimax theorem, ∃ optimal randomized

strategies for both players. The value of G ∗

n is the probability pn of capture under optimal play.

Mean capture time in Gn under optimal play is between n/pn and 2n/pn.

Yuval Peres New constructions of Kakeya and Besicovitch sets

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The n-step game G ∗

n

Define a zero sum game G ∗

n with payoff 1 to the hunter if he captures the

rabbit in the first n steps, and payoff 0 otherwise. G ∗

n is finite ⇒ By the minimax theorem, ∃ optimal randomized

strategies for both players. The value of G ∗

n is the probability pn of capture under optimal play.

Mean capture time in Gn under optimal play is between n/pn and 2n/pn. We will estimate pn, and construct a Besicovitch set of area ≍ pn, that consists of 4n triangles.

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Micah Adler, Harald R¨ acke, Naveen Sivadasan, Christian Sohler, and Berthold V¨

cking.

Randomized pursuit-evasion in graphs.

Combin. Probab. Comput., 12(3):225–244, 2003.

Combinatorics, probability and computing (Oberwolfach, 2001). Yakov Babichenko, Yuval Peres, Ron Peretz, Perla Sousi, and Peter Winkler. Hunter, Cauchy Rabbit and Optimal Kakeya Sets. Available at arXiv:1207.6389

A. S. Besicovitch.

On Kakeya’s problem and a similar one.

Math. Z., 27(1):312–320, 1928.

Roy O. Davies. Some remarks on the Kakeya problem.

Proc. Cambridge Philos. Soc., 69:417–421, 1971.

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Examples of strategies

If the rabbit chooses a random node and stays there, the hunter can sweep the cycle, so probability of capture in n steps is about 1/2.

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Examples of strategies

If the rabbit chooses a random node and stays there, the hunter can sweep the cycle, so probability of capture in n steps is about 1/2. What if the rabbit jumps to a uniform random node in each step?

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Examples of strategies

If the rabbit chooses a random node and stays there, the hunter can sweep the cycle, so probability of capture in n steps is about 1/2. What if the rabbit jumps to a uniform random node in each step? Then, for any hunter strategy, he will capture the rabbit with probability 1/n at each step, so probability of capture in n steps is 1 − (1 − 1/n)n → 1 − 1/e.

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Examples of strategies

If the rabbit chooses a random node and stays there, the hunter can sweep the cycle, so probability of capture in n steps is about 1/2. What if the rabbit jumps to a uniform random node in each step? Then, for any hunter strategy, he will capture the rabbit with probability 1/n at each step, so probability of capture in n steps is 1 − (1 − 1/n)n → 1 − 1/e. Zig-Zag hunter strategy: He starts in a random direction, then switches direction with probability 1/n at each step.

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Examples of strategies

If the rabbit chooses a random node and stays there, the hunter can sweep the cycle, so probability of capture in n steps is about 1/2. What if the rabbit jumps to a uniform random node in each step? Then, for any hunter strategy, he will capture the rabbit with probability 1/n at each step, so probability of capture in n steps is 1 − (1 − 1/n)n → 1 − 1/e. Zig-Zag hunter strategy: He starts in a random direction, then switches direction with probability 1/n at each step. Rabbit counter-strategy: From a random starting node, the rabbit walks √n steps to the right, then jumps 2√n to the left, and repeats. The probability of capture in n steps is ≍ n−1/2.

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Zig-Zag hunter strategy

time space time space

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Hunter’s optimal strategy

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Hunter’s optimal strategy

It turns out the best the hunter can do is start at a random point and continue at a random speed.

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Hunter’s optimal strategy

It turns out the best the hunter can do is start at a random point and continue at a random speed. More formally....

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Hunter’s optimal strategy

It turns out the best the hunter can do is start at a random point and continue at a random speed. More formally.... Let a,b be independent uniform on [0, 1].

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Hunter’s optimal strategy

It turns out the best the hunter can do is start at a random point and continue at a random speed. More formally.... Let a,b be independent uniform on [0, 1]. Let the position of the hunter at time t be Ht = ⌈an + bt⌉ mod n.

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Hunter’s optimal strategy

It turns out the best the hunter can do is start at a random point and continue at a random speed. More formally.... Let a,b be independent uniform on [0, 1]. Let the position of the hunter at time t be Ht = ⌈an + bt⌉ mod n. What capture time does this yield?

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Hunter’s optimal strategy

It turns out the best the hunter can do is start at a random point and continue at a random speed. More formally.... Let a,b be independent uniform on [0, 1]. Let the position of the hunter at time t be Ht = ⌈an + bt⌉ mod n. What capture time does this yield? Let Rℓ be the position of the rabbit at time ℓ and Kn the number of collisions

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Hunter’s optimal strategy

It turns out the best the hunter can do is start at a random point and continue at a random speed. More formally.... Let a,b be independent uniform on [0, 1]. Let the position of the hunter at time t be Ht = ⌈an + bt⌉ mod n. What capture time does this yield? Let Rℓ be the position of the rabbit at time ℓ and Kn the number of collisions, i.e. Kn =

n−1

i=0 1(Ri = Hi).

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Hunter’s optimal strategy

It turns out the best the hunter can do is start at a random point and continue at a random speed. More formally.... Let a,b be independent uniform on [0, 1]. Let the position of the hunter at time t be Ht = ⌈an + bt⌉ mod n. What capture time does this yield? Let Rℓ be the position of the rabbit at time ℓ and Kn the number of collisions, i.e. Kn =

n−1

i=0 1(Ri = Hi).

Use second moment method – calculate first and second moments of Kn.

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Hunter’s optimal strategy

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Hunter’s optimal strategy

We will show that P(Kn > 0)

1 log n.

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Hunter’s optimal strategy

We will show that P(Kn > 0)

1 log n.

Recall Kn = n−1

i=0 1(Ri = Hi)

hunterrabbithunterrabbithunterrrrrrrrr

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Hunter’s optimal strategy

We will show that P(Kn > 0)

1 log n.

Recall Kn = n−1

i=0 1(Ri = Hi)

hunterrabbithunterrabbithunterrrrrrrrr Ht = ⌈an + bt⌉ mod n

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Hunter’s optimal strategy

We will show that P(Kn > 0)

1 log n.

Recall Kn = n−1

i=0 1(Ri = Hi)

hunterrabbithunterrabbithunterrrrrrrrr Ht = ⌈an + bt⌉ mod n E[Kn] =

n−1

i=0

P(Hi = Ri) = 1

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 60

Hunter’s optimal strategy

We will show that P(Kn > 0)

1 log n.

Recall Kn = n−1

i=0 1(Ri = Hi)

hunterrabbithunterrabbithunterrrrrrrrr Ht = ⌈an + bt⌉ mod n E[Kn] =

n−1

i=0

P(Hi = Ri) = 1 E

K 2

n

= E[Kn] +
i=ℓ

P(Hi = Ri, Hℓ = Rℓ)

Yuval Peres New constructions of Kakeya and Besicovitch sets

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Hunter’s optimal strategy

We will show that P(Kn > 0)

1 log n.

Recall Kn = n−1

i=0 1(Ri = Hi)

hunterrabbithunterrabbithunterrrrrrrrr Ht = ⌈an + bt⌉ mod n E[Kn] =

n−1

i=0

P(Hi = Ri) = 1 E

K 2

n

= E[Kn] +
i=ℓ

P(Hi = Ri, Hℓ = Rℓ) Suffices to show E

K 2

n

log n

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 62

Hunter’s optimal strategy

We will show that P(Kn > 0)

1 log n.

Recall Kn = n−1

i=0 1(Ri = Hi)

hunterrabbithunterrabbithunterrrrrrrrr Ht = ⌈an + bt⌉ mod n E[Kn] =

n−1

i=0

P(Hi = Ri) = 1 E

K 2

n

= E[Kn] +
i=ℓ

P(Hi = Ri, Hℓ = Rℓ) Suffices to show E

K 2

n

log n

Then by Cauchy-Schwartz P(Kn > 0) ≥ E[Kn]2 E[K 2

n ]

1 log n .

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 63

Hunter’s optimal strategy

We will show that P(Kn > 0)

1 log n.

Recall Kn = n−1

i=0 1(Ri = Hi)

hunterrabbithunterrabbithunterrrrrrrrr Ht = ⌈an + bt⌉ mod n E[Kn] =

n−1

i=0

P(Hi = Ri) = 1 E

K 2

n

= E[Kn] +
i=ℓ

P(Hi = Ri, Hℓ = Rℓ) Suffices to show E

K 2

n

log n

Then by Cauchy-Schwartz P(Kn > 0) ≥ E[Kn]2 E[K 2

n ]

1 log n . Enough to prove P(Hi = Ri, Hi+j = Ri+j) 1

jn

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 64

Hunter’s optimal strategy

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 65

Hunter’s optimal strategy

Need to prove P(Hi = Ri, Hi+j = Ri+j) 1 jn .

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 66

Hunter’s optimal strategy

Need to prove P(Hi = Ri, Hi+j = Ri+j) 1 jn . Recall a, b ∼ U[0, 1]

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 67

Hunter’s optimal strategy

Need to prove P(Hi = Ri, Hi+j = Ri+j) 1 jn . This is equivalent to showing that for r, s fixed Recall a, b ∼ U[0, 1]

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 68

Hunter’s optimal strategy

Need to prove P(Hi = Ri, Hi+j = Ri+j) 1 jn . This is equivalent to showing that for r, s fixed Recall a, b ∼ U[0, 1] P(an + bi ∈ (r − 1, r], na + b(i + j) ∈ (s − 1, s]) 1 jn .

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 69

Hunter’s optimal strategy

Need to prove P(Hi = Ri, Hi+j = Ri+j) 1 jn . This is equivalent to showing that for r, s fixed Recall a, b ∼ U[0, 1] P(an + bi ∈ (r − 1, r], na + b(i + j) ∈ (s − 1, s]) 1 jn . Subtract the two constraints to get bj ∈ [s − r − 1, s − r + 1] – this has measure at most 2/j.

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 70

Hunter’s optimal strategy

Need to prove P(Hi = Ri, Hi+j = Ri+j) 1 jn . This is equivalent to showing that for r, s fixed Recall a, b ∼ U[0, 1] P(an + bi ∈ (r − 1, r], na + b(i + j) ∈ (s − 1, s]) 1 jn . Subtract the two constraints to get bj ∈ [s − r − 1, s − r + 1] – this has measure at most 2/j. After fixing b, the choices for a have measure 1/n.

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 71

Hunter’s optimal strategy

Need to prove P(Hi = Ri, Hi+j = Ri+j) 1 jn . This is equivalent to showing that for r, s fixed Recall a, b ∼ U[0, 1] P(an + bi ∈ (r − 1, r], na + b(i + j) ∈ (s − 1, s]) 1 jn . Subtract the two constraints to get bj ∈ [s − r − 1, s − r + 1] – this has measure at most 2/j. After fixing b, the choices for a have measure 1/n.

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 72

Rabbit’s optimal strategy

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 73

Rabbit’s optimal strategy

Recall Kn = n−1

i=0 1(Hi = Ri)

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 74

Rabbit’s optimal strategy

With the hunter’s strategy above rrr Recall Kn = n−1

i=0 1(Hi = Ri)

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 75

Rabbit’s optimal strategy

With the hunter’s strategy above rrr Recall Kn = n−1

i=0 1(Hi = Ri)

P(Kn > 0) 1 log n .

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 76

Rabbit’s optimal strategy

With the hunter’s strategy above rrr Recall Kn = n−1

i=0 1(Hi = Ri)

P(Kn > 0) 1 log n . This gave expected capture time at most n log n.

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 77

Rabbit’s optimal strategy

With the hunter’s strategy above rrr Recall Kn = n−1

i=0 1(Hi = Ri)

P(Kn > 0) 1 log n . This gave expected capture time at most n log n. What about the rabbit?

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 78

Rabbit’s optimal strategy

With the hunter’s strategy above rrr Recall Kn = n−1

i=0 1(Hi = Ri)

P(Kn > 0) 1 log n . This gave expected capture time at most n log n. What about the rabbit? Can he escape for time of order n log n?

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 79

Rabbit’s optimal strategy

With the hunter’s strategy above rrr Recall Kn = n−1

i=0 1(Hi = Ri)

P(Kn > 0) 1 log n . This gave expected capture time at most n log n. What about the rabbit? Can he escape for time of order n log n? Looking for a rabbit strategy with P(Kn > 0) 1 log n .

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 80

Rabbit’s optimal strategy

With the hunter’s strategy above rrr Recall Kn = n−1

i=0 1(Hi = Ri)

P(Kn > 0) 1 log n . This gave expected capture time at most n log n. What about the rabbit? Can he escape for time of order n log n? Looking for a rabbit strategy with P(Kn > 0) 1 log n . Extend the strategies until time 2n and define K2n analogously.

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 81

Rabbit’s optimal strategy

With the hunter’s strategy above rrr Recall Kn = n−1

i=0 1(Hi = Ri)

P(Kn > 0) 1 log n . This gave expected capture time at most n log n. What about the rabbit? Can he escape for time of order n log n? Looking for a rabbit strategy with P(Kn > 0) 1 log n . Extend the strategies until time 2n and define K2n analogously. Obviously P(Kn > 0) ≤ E[K2n] E[K2n | Kn > 0]

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 82

Rabbit’s optimal strategy

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 83

Rabbit’s optimal strategy

If the rabbit starts at a uniform point and the jumps are independent, then

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 84

Rabbit’s optimal strategy

If the rabbit starts at a uniform point and the jumps are independent, then E[K2n] = 2 rrrrrr Recall K2n =

2n−1

i=0 1(Hi = Ri)

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 85

Rabbit’s optimal strategy

If the rabbit starts at a uniform point and the jumps are independent, then E[K2n] = 2 rrrrrr Recall K2n =

2n−1

i=0 1(Hi = Ri)

Idea: Need to make E[K2n | Kn > 0] “big” so P(Kn > 0) ≤ (log n)−1.

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 86

Rabbit’s optimal strategy

If the rabbit starts at a uniform point and the jumps are independent, then E[K2n] = 2 rrrrrr Recall K2n =

2n−1

i=0 1(Hi = Ri)

Idea: Need to make E[K2n | Kn > 0] “big” so P(Kn > 0) ≤ (log n)−1. This means that given the rabbit and hunter collided, we want them to collide “a lot”. The hunter can only move to neighbours or stay put.

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 87

Rabbit’s optimal strategy

If the rabbit starts at a uniform point and the jumps are independent, then E[K2n] = 2 rrrrrr Recall K2n =

2n−1

i=0 1(Hi = Ri)

Idea: Need to make E[K2n | Kn > 0] “big” so P(Kn > 0) ≤ (log n)−1. This means that given the rabbit and hunter collided, we want them to collide “a lot”. The hunter can only move to neighbours or stay put. So the rabbit should also choose a distribution for the jumps that favors short distances, yet grows linearly in time. This suggests a Cauchy random walk.

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 88

Cauchy Rabbit

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 89

Cauchy Rabbit

By time i the hunter can only be in the set {−i mod n, . . . , i mod n}. We are looking for a distribution for the rabbit so that

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 90

Cauchy Rabbit

By time i the hunter can only be in the set {−i mod n, . . . , i mod n}. We are looking for a distribution for the rabbit so that P(Ri = ℓ) 1 i for ℓ ∈ {−i mod n, . . . , i mod n}.

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 91

Cauchy Rabbit

By time i the hunter can only be in the set {−i mod n, . . . , i mod n}. We are looking for a distribution for the rabbit so that P(Ri = ℓ) 1 i for ℓ ∈ {−i mod n, . . . , i mod n}. Then by the Markov property E[K2n | Kn > 0] ≥

n−1

i=0

P0(Hi = Ri) log n.

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 92

Cauchy Rabbit

By time i the hunter can only be in the set {−i mod n, . . . , i mod n}. We are looking for a distribution for the rabbit so that P(Ri = ℓ) 1 i for ℓ ∈ {−i mod n, . . . , i mod n}. Then by the Markov property E[K2n | Kn > 0] ≥

n−1

i=0

P0(Hi = Ri) log n. Intuition: If X1, . . . are i.i.d. Cauchy random variables, i.e. with density (π(1 + x2))−1, then X1 + . . . + Xn is spread over (−n, n) and with roughly uniform distribution.

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 93

Cauchy Rabbit

By time i the hunter can only be in the set {−i mod n, . . . , i mod n}. We are looking for a distribution for the rabbit so that P(Ri = ℓ) 1 i for ℓ ∈ {−i mod n, . . . , i mod n}. Then by the Markov property E[K2n | Kn > 0] ≥

n−1

i=0

P0(Hi = Ri) log n. Intuition: If X1, . . . are i.i.d. Cauchy random variables, i.e. with density (π(1 + x2))−1, then X1 + . . . + Xn is spread over (−n, n) and with roughly uniform distribution. This is what we want- But in the discrete setting...

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 94

Cauchy Rabbit

The Cauchy distribution can be embedded in planar Brownian motion.

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 95

Cauchy Rabbit

The Cauchy distribution can be embedded in planar Brownian motion. Let’s imitate that in the discrete setting:

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 96

Cauchy Rabbit

The Cauchy distribution can be embedded in planar Brownian motion. Let’s imitate that in the discrete setting: Let (Xt, Yt)t be a simple random walk in Z2.Define hitting times Ti = inf{t ≥ 0 : Yt = i} and set Ri = XTi modn.

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 97

Cauchy Rabbit

The Cauchy distribution can be embedded in planar Brownian motion. Let’s imitate that in the discrete setting: Let (Xt, Yt)t be a simple random walk in Z2.Define hitting times Ti = inf{t ≥ 0 : Yt = i} and set Ri = XTi modn.

)

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 98

Cauchy Rabbit

The Cauchy distribution can be embedded in planar Brownian motion. Let’s imitate that in the discrete setting: Let (Xt, Yt)t be a simple random walk in Z2.Define hitting times Ti = inf{t ≥ 0 : Yt = i} and set Ri = XTi modn. With probability 1/4, SRW exits the square via the top side.

)

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 99

Cauchy Rabbit

The Cauchy distribution can be embedded in planar Brownian motion. Let’s imitate that in the discrete setting: Let (Xt, Yt)t be a simple random walk in Z2.Define hitting times Ti = inf{t ≥ 0 : Yt = i} and set Ri = XTi modn. With probability 1/4, SRW exits the square via the top side. Of the 2i + 1 nodes on the top, the middle node is the most likely hitting point: subdivide all edges, and condition on the (even) number of horizontal steps until height i is reached; the horizontal displacement is a shifted binomial, so the mode is the mean.

)

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 100

Cauchy Rabbit

The Cauchy distribution can be embedded in planar Brownian motion. Let’s imitate that in the discrete setting: Let (Xt, Yt)t be a simple random walk in Z2.Define hitting times Ti = inf{t ≥ 0 : Yt = i} and set Ri = XTi modn. With probability 1/4, SRW exits the square via the top side. Of the 2i + 1 nodes on the top, the middle node is the most likely hitting point: subdivide all edges, and condition on the (even) number of horizontal steps until height i is reached; the horizontal displacement is a shifted binomial, so the mode is the mean. Thus the hitting probability at (0, i) is at least 1/(8i + 4).

)

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 101

Cauchy Rabbit

time space

(k,k) (k,k) (k,k) (k,k) (0,0) (k,i) (k,l)

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 102

Cauchy Rabbit

Suppose 0 < k < i.

time space

(k,k) (k,k) (k,k) (k,k) (0,0) (k,i) (k,l)

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 103

Cauchy Rabbit

Suppose 0 < k < i. With probability 1/4, SRW exits the square [−k, k]2 via the right side.

time space

(k,k) (k,k) (k,k) (k,k) (0,0) (k,i) (k,l)

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 104

Cauchy Rabbit

Suppose 0 < k < i. With probability 1/4, SRW exits the square [−k, k]2 via the right side. Repeating the previous argument, the hitting probability at (k, i) is at least c/i.

time space

(k,k) (k,k) (k,k) (k,k) (0,0) (k,i) (k,l)

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 105

Rabbit and Hunter construct Besicovitch sets

Let (Rt)t be a rabbit strategy. Extend it to real times as a step function.

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 106

Rabbit and Hunter construct Besicovitch sets

Let (Rt)t be a rabbit strategy. Extend it to real times as a step function. Let a be uniform in [−1, 1] and b uniform in [0, 1] and Ht = an + bt. There is a collision at time t ∈ [0, n) if Rt = Ht.

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 107

Rabbit and Hunter construct Besicovitch sets

Let (Rt)t be a rabbit strategy. Extend it to real times as a step function. Let a be uniform in [−1, 1] and b uniform in [0, 1] and Ht = an + bt. There is a collision at time t ∈ [0, n) if Rt = Ht. What is the chance there is a collision in [m, m + 1)?

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 108

Rabbit and Hunter construct Besicovitch sets

Let (Rt)t be a rabbit strategy. Extend it to real times as a step function. Let a be uniform in [−1, 1] and b uniform in [0, 1] and Ht = an + bt. There is a collision at time t ∈ [0, n) if Rt = Ht. What is the chance there is a collision in [m, m + 1)? It is P(an + bm ≤ Rm < an + b(m + 1)), which is half the area of the triangle

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 109

Rabbit and Hunter construct Besicovitch sets

Let (Rt)t be a rabbit strategy. Extend it to real times as a step function. Let a be uniform in [−1, 1] and b uniform in [0, 1] and Ht = an + bt. There is a collision at time t ∈ [0, n) if Rt = Ht. What is the chance there is a collision in [m, m + 1)? It is P(an + bm ≤ Rm < an + b(m + 1)), which is half the area of the triangle

Rm n

a b 1

1

−1

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 110

Rabbit and Hunter construct Besicovitch sets

Let (Rt)t be a rabbit strategy. Extend it to real times as a step function. Let a be uniform in [−1, 1] and b uniform in [0, 1] and Ht = an + bt. There is a collision at time t ∈ [0, n) if Rt = Ht. What is the chance there is a collision in [m, m + 1)? It is P(an + bm ≤ Rm < an + b(m + 1)), which is half the area of the triangle

Rm n

a b 1

1

−1

an + bm = Rm an + b(m + 1) = Rm

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 111

Rabbit and Hunter construct Besicovitch sets

Hence the probability of collision in [0, n) is half the area of the union of all such triangles, which are translates of T1 T2 Tn Tn−1 1

1 n 1 n

. . .

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 112

Rabbit and Hunter construct Besicovitch sets

Hence the probability of collision in [0, n) is half the area of the union of all such triangles, which are translates of T1 T2 Tn Tn−1 1

1 n 1 n

. . . In these triangles we can find a unit segment in all directions that have an angle in [0, π/4]

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 113

Rabbit and Hunter construct Besicovitch sets

If the rabbit employs the Cauchy strategy, then P(collision in the first n steps) 1 log n .

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 114

Rabbit and Hunter construct Besicovitch sets

If the rabbit employs the Cauchy strategy, then P(collision in the first n steps) 1 log n . Hence, this gives a set of triangles with area of order at most 1/ log n.

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 115

Rabbit and Hunter construct Besicovitch sets

If the rabbit employs the Cauchy strategy, then P(collision in the first n steps) 1 log n . Hence, this gives a set of triangles with area of order at most 1/ log n. Simulation generated with n = 32

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 116

Besicovitch sets from the Cauchy process

Motivated by the Cauchy strategy, let’s see a continuum analog of the probabilistic Kakeya construction of the hunter and rabbit.

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 117

Besicovitch sets from the Cauchy process

Motivated by the Cauchy strategy, let’s see a continuum analog of the probabilistic Kakeya construction of the hunter and rabbit. Let (Xt)t be a Cauchy process, i.e. Xt+s − Xt has the same law as tX1 and X1 has the Cauchy distribution (density given by (π(1 + x2))−1).

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 118

Besicovitch sets from the Cauchy process

Motivated by the Cauchy strategy, let’s see a continuum analog of the probabilistic Kakeya construction of the hunter and rabbit. Let (Xt)t be a Cauchy process, i.e. Xt+s − Xt has the same law as tX1 and X1 has the Cauchy distribution (density given by (π(1 + x2))−1).Set Λ = {(a, Xt + at) : a, t ∈ [0, 1]}.

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 119

Besicovitch sets from the Cauchy process

Motivated by the Cauchy strategy, let’s see a continuum analog of the probabilistic Kakeya construction of the hunter and rabbit. Let (Xt)t be a Cauchy process, i.e. Xt+s − Xt has the same law as tX1 and X1 has the Cauchy distribution (density given by (π(1 + x2))−1).Set Λ = {(a, Xt + at) : a, t ∈ [0, 1]}. Λ is a quarter of a Kakeya set – it contains all directions from 0 up to 45◦

degrees. Take four rotated copies of Λ to obtain a Kakeya set.

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 120

Besicovitch sets from the Cauchy process

Motivated by the Cauchy strategy, let’s see a continuum analog of the probabilistic Kakeya construction of the hunter and rabbit. Let (Xt)t be a Cauchy process, i.e. Xt+s − Xt has the same law as tX1 and X1 has the Cauchy distribution (density given by (π(1 + x2))−1).Set Λ = {(a, Xt + at) : a, t ∈ [0, 1]}. Λ is a quarter of a Kakeya set – it contains all directions from 0 up to 45◦

degrees. Take four rotated copies of Λ to obtain a Kakeya set.

Λ is an optimal Kakeya set!

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 121

Besicovitch sets from the Cauchy process

Motivated by the Cauchy strategy, let’s see a continuum analog of the probabilistic Kakeya construction of the hunter and rabbit. Let (Xt)t be a Cauchy process, i.e. Xt+s − Xt has the same law as tX1 and X1 has the Cauchy distribution (density given by (π(1 + x2))−1).Set Λ = {(a, Xt + at) : a, t ∈ [0, 1]}. Λ is a quarter of a Kakeya set – it contains all directions from 0 up to 45◦

degrees. Take four rotated copies of Λ to obtain a Kakeya set.

Λ is an optimal Kakeya set! Leb(Λ) = 0 and most importantly the ε-neighbourhood satisfies almost surely Leb(Λ(ε)) ≍ 1 | log ε|

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 122

Ideas of the proof

Let’s forget about the term at.

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 123

Ideas of the proof

Let’s forget about the term at. E[Leb(∪s≤1B(Xs, ε))] =

R

P

τB(x,ε) ≤ 1
dx.

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 124

Ideas of the proof

Let’s forget about the term at. E[Leb(∪s≤1B(Xs, ε))] =

R

P

τB(x,ε) ≤ 1
dx.

Defining Zx = 1

0 1(Xs ∈ B(x, ε)) ds and

Zx = 2

0 1(Xs ∈ B(x, ε)) ds,

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 125

Ideas of the proof

Let’s forget about the term at. E[Leb(∪s≤1B(Xs, ε))] =

R

P

τB(x,ε) ≤ 1
dx.

Defining Zx = 1

0 1(Xs ∈ B(x, ε)) ds and

Zx = 2

0 1(Xs ∈ B(x, ε)) ds, we have

P

τB(x,ε) ≤ 1
≤

E

Zx
E
Zx
Zx > 0

.

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 126

Ideas of the proof

Let’s forget about the term at. E[Leb(∪s≤1B(Xs, ε))] =

R

P

τB(x,ε) ≤ 1
dx.

Defining Zx = 1

0 1(Xs ∈ B(x, ε)) ds and

Zx = 2

0 1(Xs ∈ B(x, ε)) ds, we have

P

τB(x,ε) ≤ 1
≤

E

Zx
E
Zx
Zx > 0

. Using that Xs has the same law as sX1, we get that E

Zx
Zx > 0
Yuval Peres

New constructions of Kakeya and Besicovitch sets

SLIDE 127

Ideas of the proof

Let’s forget about the term at. E[Leb(∪s≤1B(Xs, ε))] =

R

P

τB(x,ε) ≤ 1
dx.

Defining Zx = 1

0 1(Xs ∈ B(x, ε)) ds and

Zx = 2

0 1(Xs ∈ B(x, ε)) ds, we have

P

τB(x,ε) ≤ 1
≤

E

Zx
E
Zx
Zx > 0

. Using that Xs has the same law as sX1, we get that E

Zx
Zx > 0
≥

min

y∈B(x,ε)

1

B( x

s − y s , ε s )

1 π(1 + z2) dz ds

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 128

Ideas of the proof

Let’s forget about the term at. E[Leb(∪s≤1B(Xs, ε))] =

R

P

τB(x,ε) ≤ 1
dx.

Defining Zx = 1

0 1(Xs ∈ B(x, ε)) ds and

Zx = 2

0 1(Xs ∈ B(x, ε)) ds, we have

P

τB(x,ε) ≤ 1
≤

E

Zx
E
Zx
Zx > 0

. Using that Xs has the same law as sX1, we get that E

Zx
Zx > 0
≥

min

y∈B(x,ε)

1

B( x

s − y s , ε s )

1 π(1 + z2) dz ds ≥ cε | log ε|.

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 129

Ideas of the proof

Let’s forget about the term at. E[Leb(∪s≤1B(Xs, ε))] =

R

P

τB(x,ε) ≤ 1
dx.

Defining Zx = 1

0 1(Xs ∈ B(x, ε)) ds and

Zx = 2

0 1(Xs ∈ B(x, ε)) ds, we have

P

τB(x,ε) ≤ 1
≤

E

Zx
E
Zx
Zx > 0

. Using that Xs has the same law as sX1, we get that E

Zx
Zx > 0
≥

min

y∈B(x,ε)

1

B( x

s − y s , ε s )

1 π(1 + z2) dz ds ≥ cε | log ε|. Hence E[Leb(∪s≤1B(Xs, ε))] ≤ 1 | log ε|

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 130

Kakeya sets – Open problems

Keich in 1999 showed there is no Besicovitch set which is a union of n triangles with area of smaller order than 1/ log n. Bourgain and Cordoba earlier noted that the ε neighborhood of any Kakeya set has area at least 1/|logε|.

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 131

Kakeya sets – Open problems

Keich in 1999 showed there is no Besicovitch set which is a union of n triangles with area of smaller order than 1/ log n. Bourgain and Cordoba earlier noted that the ε neighborhood of any Kakeya set has area at least 1/|logε|. So the random construction is optimal.

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 132

Kakeya sets – Open problems

Keich in 1999 showed there is no Besicovitch set which is a union of n triangles with area of smaller order than 1/ log n. Bourgain and Cordoba earlier noted that the ε neighborhood of any Kakeya set has area at least 1/|logε|. So the random construction is optimal. Davies in 1971 showed that Besicovitch sets in the plane have Hausdorff dimension equal to 2.

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 133

Kakeya sets – Open problems

Keich in 1999 showed there is no Besicovitch set which is a union of n triangles with area of smaller order than 1/ log n. Bourgain and Cordoba earlier noted that the ε neighborhood of any Kakeya set has area at least 1/|logε|. So the random construction is optimal. Davies in 1971 showed that Besicovitch sets in the plane have Hausdorff dimension equal to 2. It is a major open problem whether Besicovitch sets in dimensions d > 2 have Hausdorff dimension equal to d.

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 134

Kakeya sets – Open problems

Keich in 1999 showed there is no Besicovitch set which is a union of n triangles with area of smaller order than 1/ log n. Bourgain and Cordoba earlier noted that the ε neighborhood of any Kakeya set has area at least 1/|logε|. So the random construction is optimal. Davies in 1971 showed that Besicovitch sets in the plane have Hausdorff dimension equal to 2. It is a major open problem whether Besicovitch sets in dimensions d > 2 have Hausdorff dimension equal to d.

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 135

Cauchy process

The Cauchy process can be embedded in planar Brownian motion.

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 136

Cauchy process

The Cauchy process can be embedded in planar Brownian motion.

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 137

Cauchy process

The Cauchy process can be embedded in planar Brownian motion. (Xt, t) t

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 138

A construction from Bishop-P., Fractals in Probability and Analysis (cf E. Sawyer (1987))

Theorem (Besicovitch 1919, 1928) There is a set of zero area in R2 that contains a unit line segment in every direction. Proof: Consider the sequence {ak}∞

k=1 =

0, 1, 1

2, 0, 1 4, 2 4, 3 4, 1, 7 8, 6 8, 5 8, . . .

,

i.e., a1 = 0 and for k ∈ [2n, 2n+1), ak =

k2−n − 1

if n is even 2 − k2−n if n is odd Set g(t) = t − ⌊t⌋, fk(t) =

k

j=2

aj−1 − aj 2j g(2jt), and f (t) = lim

k→∞ fk(t).

By telescoping, f ′

k(t) = −ak on each component of U = [0, 1] \ 2−kZ.

Yuval Peres New constructions of Kakeya and Besicovitch sets

SLIDE 139

Let K = {(a, f (t) + at) : a, t ∈ [0, 1]}.

Fixing t shows K contains unit segments of all slopes in [0, 1], so a union

f four rotations of K contains unit segments of all slopes.

We need to show K has zero area. Given a ∈ [0, 1] and n ≥ 1, find k ∈ [2n, 2n+1] so that |a − ak| ≤ 2−n. Then fk(t) + at is piecewise linear with |slopes| ≤ 2−n on the 2k components of U = [0, 1] \ 2−kZ. Hence this function maps each such component I into an interval of length at most 2−n|I| = 2−n−k. Also, |f (t) − fk(t)| ≤

∞

j=k+1

|aj−1 − aj| 2j g(2jt) ≤ 2−n

∞

j=k+1

2−j = 2−n−k . Thus f (t) + at maps each component I of U into an interval of length ≤ 3 · 2−n−k Thus every vertical slice {t : (a, t) ∈ K} has length zero, so by Fubini’s Theorem, K has zero area.

Yuval Peres New constructions of Kakeya and Besicovitch sets