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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 1 Microsoft Research Yuval Peres New

  1. Definition of the game G n 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 Z n . When does the game end? At “capture time”, when the hunter and the rabbit occupy the same location in Z n at the same time. Goals rrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr hhhhhhhhhhhhh Yuval Peres New constructions of Kakeya and Besicovitch sets

  2. Definition of the game G n 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 Z n . When does the game end? At “capture time”, when the hunter and the rabbit occupy the same location in Z n at the same time. Goals Hunter: Minimize “capture time” rrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr hhhhhhhhhhhhh Yuval Peres New constructions of Kakeya and Besicovitch sets

  3. Definition of the game G n 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 Z n . When does the game end? At “capture time”, when the hunter and the rabbit occupy the same location in Z n 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

  4. 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

  5. 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

  6. 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 p n of capture under optimal play. Yuval Peres New constructions of Kakeya and Besicovitch sets

  7. 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 p n of capture under optimal play. Mean capture time in G n under optimal play is between n / p n and 2 n / p n . Yuval Peres New constructions of Kakeya and Besicovitch sets

  8. 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 p n of capture under optimal play. Mean capture time in G n under optimal play is between n / p n and 2 n / p n . We will estimate p n , and construct a Besicovitch set of area ≍ p n , that consists of 4 n triangles. Yuval Peres New constructions of Kakeya and Besicovitch sets

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

  10. 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

  11. 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

  12. 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

  13. 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

  14. 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

  15. Zig-Zag hunter strategy time time space space Yuval Peres New constructions of Kakeya and Besicovitch sets

  16. Hunter’s optimal strategy Yuval Peres New constructions of Kakeya and Besicovitch sets

  17. 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

  18. 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

  19. 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

  20. 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 H t = ⌈ an + bt ⌉ mod n . Yuval Peres New constructions of Kakeya and Besicovitch sets

  21. 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 H t = ⌈ an + bt ⌉ mod n . What capture time does this yield? Yuval Peres New constructions of Kakeya and Besicovitch sets

  22. 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 H t = ⌈ an + bt ⌉ mod n . What capture time does this yield? Let R ℓ be the position of the rabbit at time ℓ and K n the number of collisions Yuval Peres New constructions of Kakeya and Besicovitch sets

  23. 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 H t = ⌈ an + bt ⌉ mod n . What capture time does this yield? Let R ℓ be the position of the rabbit at time ℓ and K n the number of collisions , i.e. n − 1 � K n = i =0 1 ( R i = H i ) . Yuval Peres New constructions of Kakeya and Besicovitch sets

  24. 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 H t = ⌈ an + bt ⌉ mod n . What capture time does this yield? Let R ℓ be the position of the rabbit at time ℓ and K n the number of collisions , i.e. n − 1 � K n = i =0 1 ( R i = H i ) . Use second moment method – calculate first and second moments of K n . Yuval Peres New constructions of Kakeya and Besicovitch sets

  25. Hunter’s optimal strategy Yuval Peres New constructions of Kakeya and Besicovitch sets

  26. Hunter’s optimal strategy 1 We will show that P ( K n > 0) � log n . Yuval Peres New constructions of Kakeya and Besicovitch sets

  27. Hunter’s optimal strategy Recall K n = � n − 1 1 We will show that P ( K n > 0) � log n . i =0 1 ( R i = H i ) hunterrabbithunterrabbithunterrrrrrrrr Yuval Peres New constructions of Kakeya and Besicovitch sets

  28. Hunter’s optimal strategy Recall K n = � n − 1 1 We will show that P ( K n > 0) � log n . i =0 1 ( R i = H i ) hunterrabbithunterrabbithunterrrrrrrrr H t = ⌈ an + bt ⌉ mod n Yuval Peres New constructions of Kakeya and Besicovitch sets

  29. Hunter’s optimal strategy Recall K n = � n − 1 1 We will show that P ( K n > 0) � log n . i =0 1 ( R i = H i ) hunterrabbithunterrabbithunterrrrrrrrr H t = ⌈ an + bt ⌉ mod n n − 1 � E [ K n ] = P ( H i = R i ) = 1 i =0 Yuval Peres New constructions of Kakeya and Besicovitch sets

  30. Hunter’s optimal strategy Recall K n = � n − 1 1 We will show that P ( K n > 0) � log n . i =0 1 ( R i = H i ) hunterrabbithunterrabbithunterrrrrrrrr H t = ⌈ an + bt ⌉ mod n n − 1 � E [ K n ] = P ( H i = R i ) = 1 i =0 � � � K 2 = E [ K n ] + P ( H i = R i , H ℓ = R ℓ ) E n i � = ℓ Yuval Peres New constructions of Kakeya and Besicovitch sets

  31. Hunter’s optimal strategy Recall K n = � n − 1 1 We will show that P ( K n > 0) � log n . i =0 1 ( R i = H i ) hunterrabbithunterrabbithunterrrrrrrrr H t = ⌈ an + bt ⌉ mod n n − 1 � E [ K n ] = P ( H i = R i ) = 1 i =0 � � � K 2 = E [ K n ] + P ( H i = R i , H ℓ = R ℓ ) E n i � = ℓ � � K 2 Suffices to show � log n E n Yuval Peres New constructions of Kakeya and Besicovitch sets

  32. Hunter’s optimal strategy Recall K n = � n − 1 1 We will show that P ( K n > 0) � log n . i =0 1 ( R i = H i ) hunterrabbithunterrabbithunterrrrrrrrr H t = ⌈ an + bt ⌉ mod n n − 1 � E [ K n ] = P ( H i = R i ) = 1 i =0 � � � K 2 = E [ K n ] + P ( H i = R i , H ℓ = R ℓ ) E n i � = ℓ � � K 2 Suffices to show � log n E n Then by Cauchy-Schwartz P ( K n > 0) ≥ E [ K n ] 2 1 n ] � log n . E [ K 2 Yuval Peres New constructions of Kakeya and Besicovitch sets

  33. Hunter’s optimal strategy Recall K n = � n − 1 1 We will show that P ( K n > 0) � log n . i =0 1 ( R i = H i ) hunterrabbithunterrabbithunterrrrrrrrr H t = ⌈ an + bt ⌉ mod n n − 1 � E [ K n ] = P ( H i = R i ) = 1 i =0 � � � K 2 = E [ K n ] + P ( H i = R i , H ℓ = R ℓ ) E n i � = ℓ � � K 2 Suffices to show � log n E n Then by Cauchy-Schwartz P ( K n > 0) ≥ E [ K n ] 2 1 n ] � log n . E [ K 2 P ( H i = R i , H i + j = R i + j ) � 1 Enough to prove jn Yuval Peres New constructions of Kakeya and Besicovitch sets

  34. Hunter’s optimal strategy Yuval Peres New constructions of Kakeya and Besicovitch sets

  35. Hunter’s optimal strategy Need to prove P ( H i = R i , H i + j = R i + j ) � 1 jn . Yuval Peres New constructions of Kakeya and Besicovitch sets

  36. Hunter’s optimal strategy Need to prove P ( H i = R i , H i + j = R i + j ) � 1 jn . Recall a , b ∼ U [0 , 1] Yuval Peres New constructions of Kakeya and Besicovitch sets

  37. Hunter’s optimal strategy Need to prove P ( H i = R i , H i + j = R i + 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

  38. Hunter’s optimal strategy Need to prove P ( H i = R i , H i + j = R i + 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

  39. Hunter’s optimal strategy Need to prove P ( H i = R i , H i + j = R i + 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

  40. Hunter’s optimal strategy Need to prove P ( H i = R i , H i + j = R i + 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

  41. Hunter’s optimal strategy Need to prove P ( H i = R i , H i + j = R i + 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

  42. Rabbit’s optimal strategy Yuval Peres New constructions of Kakeya and Besicovitch sets

  43. Rabbit’s optimal strategy Recall K n = � n − 1 i =0 1 ( H i = R i ) Yuval Peres New constructions of Kakeya and Besicovitch sets

  44. Rabbit’s optimal strategy With the hunter’s strategy above rrr Recall K n = � n − 1 i =0 1 ( H i = R i ) Yuval Peres New constructions of Kakeya and Besicovitch sets

  45. Rabbit’s optimal strategy With the hunter’s strategy above rrr Recall K n = � n − 1 i =0 1 ( H i = R i ) 1 P ( K n > 0) � log n . Yuval Peres New constructions of Kakeya and Besicovitch sets

  46. Rabbit’s optimal strategy With the hunter’s strategy above rrr Recall K n = � n − 1 i =0 1 ( H i = R i ) 1 P ( K n > 0) � log n . This gave expected capture time at most n log n. Yuval Peres New constructions of Kakeya and Besicovitch sets

  47. Rabbit’s optimal strategy With the hunter’s strategy above rrr Recall K n = � n − 1 i =0 1 ( H i = R i ) 1 P ( K n > 0) � 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

  48. Rabbit’s optimal strategy With the hunter’s strategy above rrr Recall K n = � n − 1 i =0 1 ( H i = R i ) 1 P ( K n > 0) � 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

  49. Rabbit’s optimal strategy With the hunter’s strategy above rrr Recall K n = � n − 1 i =0 1 ( H i = R i ) 1 P ( K n > 0) � 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 1 P ( K n > 0) � log n . Yuval Peres New constructions of Kakeya and Besicovitch sets

  50. Rabbit’s optimal strategy With the hunter’s strategy above rrr Recall K n = � n − 1 i =0 1 ( H i = R i ) 1 P ( K n > 0) � 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 1 P ( K n > 0) � log n . Extend the strategies until time 2 n and define K 2 n analogously. Yuval Peres New constructions of Kakeya and Besicovitch sets

  51. Rabbit’s optimal strategy With the hunter’s strategy above rrr Recall K n = � n − 1 i =0 1 ( H i = R i ) 1 P ( K n > 0) � 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 1 P ( K n > 0) � log n . Extend the strategies until time 2 n and define K 2 n analogously. Obviously E [ K 2 n ] P ( K n > 0) ≤ E [ K 2 n | K n > 0] Yuval Peres New constructions of Kakeya and Besicovitch sets

  52. Rabbit’s optimal strategy Yuval Peres New constructions of Kakeya and Besicovitch sets

  53. 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

  54. Rabbit’s optimal strategy If the rabbit starts at a uniform point and the jumps are independent, then 2 n − 1 � E [ K 2 n ] = 2 rrrrrr Recall K 2 n = i =0 1 ( H i = R i ) Yuval Peres New constructions of Kakeya and Besicovitch sets

  55. Rabbit’s optimal strategy If the rabbit starts at a uniform point and the jumps are independent, then 2 n − 1 � E [ K 2 n ] = 2 rrrrrr Recall K 2 n = i =0 1 ( H i = R i ) Idea: Need to make E [ K 2 n | K n > 0] “big” so P ( K n > 0) ≤ (log n ) − 1 . Yuval Peres New constructions of Kakeya and Besicovitch sets

  56. Rabbit’s optimal strategy If the rabbit starts at a uniform point and the jumps are independent, then 2 n − 1 � E [ K 2 n ] = 2 rrrrrr Recall K 2 n = i =0 1 ( H i = R i ) Idea: Need to make E [ K 2 n | K n > 0] “big” so P ( K n > 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

  57. Rabbit’s optimal strategy If the rabbit starts at a uniform point and the jumps are independent, then 2 n − 1 � E [ K 2 n ] = 2 rrrrrr Recall K 2 n = i =0 1 ( H i = R i ) Idea: Need to make E [ K 2 n | K n > 0] “big” so P ( K n > 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

  58. Cauchy Rabbit Yuval Peres New constructions of Kakeya and Besicovitch sets

  59. 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

  60. 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 ( R i = ℓ ) � 1 for ℓ ∈ {− i mod n , . . . , i mod n } . i Yuval Peres New constructions of Kakeya and Besicovitch sets

  61. 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 ( R i = ℓ ) � 1 for ℓ ∈ {− i mod n , . . . , i mod n } . i Then by the Markov property n − 1 � E [ K 2 n | K n > 0] ≥ P 0 ( H i = R i ) � log n . i =0 Yuval Peres New constructions of Kakeya and Besicovitch sets

  62. 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 ( R i = ℓ ) � 1 for ℓ ∈ {− i mod n , . . . , i mod n } . i Then by the Markov property � n − 1 E [ K 2 n | K n > 0] ≥ P 0 ( H i = R i ) � log n . i =0 Intuition: If X 1 , . . . are i.i.d. Cauchy random variables, i.e. with density ( π (1 + x 2 )) − 1 , then X 1 + . . . + X n is spread over ( − n , n ) and with roughly uniform distribution . Yuval Peres New constructions of Kakeya and Besicovitch sets

  63. 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 ( R i = ℓ ) � 1 for ℓ ∈ {− i mod n , . . . , i mod n } . i Then by the Markov property � n − 1 E [ K 2 n | K n > 0] ≥ P 0 ( H i = R i ) � log n . i =0 Intuition: If X 1 , . . . are i.i.d. Cauchy random variables, i.e. with density ( π (1 + x 2 )) − 1 , then X 1 + . . . + X n 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

  64. Cauchy Rabbit The Cauchy distribution can be embedded in planar Brownian motion. Yuval Peres New constructions of Kakeya and Besicovitch sets

  65. 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

  66. Cauchy Rabbit The Cauchy distribution can be embedded in planar Brownian motion. Let’s imitate that in the discrete setting: Let ( X t , Y t ) t be a simple random walk in Z 2 .Define hitting times T i = inf { t ≥ 0 : Y t = i } and set R i = X T i mod n . Yuval Peres New constructions of Kakeya and Besicovitch sets

  67. Cauchy Rabbit The Cauchy distribution can be embedded in planar Brownian motion. Let’s imitate that in the discrete setting: Let ( X t , Y t ) t be a simple random walk in Z 2 .Define hitting times T i = inf { t ≥ 0 : Y t = i } and set R i = X T i mod n . ) Yuval Peres New constructions of Kakeya and Besicovitch sets

  68. Cauchy Rabbit The Cauchy distribution can be embedded in planar Brownian motion. Let’s imitate that in the discrete setting: Let ( X t , Y t ) t be a simple random walk in Z 2 .Define hitting times T i = inf { t ≥ 0 : Y t = i } and set R i = X T i mod n . With probability 1 / 4, SRW exits the square via the top side. ) Yuval Peres New constructions of Kakeya and Besicovitch sets

  69. Cauchy Rabbit The Cauchy distribution can be embedded in planar Brownian motion. Let’s imitate that in the discrete setting: Let ( X t , Y t ) t be a simple random walk in Z 2 .Define hitting times T i = inf { t ≥ 0 : Y t = i } and set R i = X T i mod n . With probability 1 / 4, SRW exits the square via the top side. Of the 2 i + 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

  70. Cauchy Rabbit The Cauchy distribution can be embedded in planar Brownian motion. Let’s imitate that in the discrete setting: Let ( X t , Y t ) t be a simple random walk in Z 2 .Define hitting times T i = inf { t ≥ 0 : Y t = i } and set R i = X T i mod n . With probability 1 / 4, SRW exits the square via the top side. Of the 2 i + 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 / (8 i + 4). Yuval Peres New constructions of Kakeya and Besicovitch sets

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