Fubini type results for Hausdorff dimension Tam as Keleti with - - PowerPoint PPT Presentation
Fubini type results for Hausdorff dimension Tam as Keleti with Korn elia H era and Andr as M ath e E otv os Lor and University, Budapest Warwick, 12 July 2017 Tam as Keleti (Budapest) with Korn elia H era
Tam´ as Keleti with Korn´ elia H´ era and Andr´ as M´ ath´ e
E¨
and University, Budapest
Warwick, 12 July 2017
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 1 / 14
Let n ≥ 2, E ⊂ Rn, v ∈ Sn−1 and V = v ⊥ ∈ G(n, n − 1).
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 2 / 14
Let n ≥ 2, E ⊂ Rn, v ∈ Sn−1 and V = v ⊥ ∈ G(n, n − 1).
If {t ∈ R : dim(E ∩ (V + tv)) ≥ s} has positive Lebesgue measure then dim E ≥ s + 1.
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 2 / 14
Let n ≥ 2, E ⊂ Rn, v ∈ Sn−1 and V = v ⊥ ∈ G(n, n − 1).
If {t ∈ R : dim(E ∩ (V + tv)) ≥ s} has positive Lebesgue measure then dim E ≥ s + 1.
If dim(E ∩ (V + tv)) ≤ s for a.e. t ∈ R then dim E ≤ s + 1.
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 2 / 14
Let n ≥ 2, E ⊂ Rn, v ∈ Sn−1 and V = v ⊥ ∈ G(n, n − 1).
If {t ∈ R : dim(E ∩ (V + tv)) ≥ s} has positive Lebesgue measure then dim E ≥ s + 1.
If dim(E ∩ (V + tv)) ≤ s for a.e. t ∈ R then dim E ≤ s + 1. Combining the above inequalities we would get:
dim E = s + 1, where s is the essential supremum of dim(E ∩ (V + tv)) (t ∈ R).
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 2 / 14
Let n ≥ 2, E ⊂ Rn, v ∈ Sn−1 and V = v ⊥ ∈ G(n, n − 1).
If dim(E ∩ (V + tv)) ≤ s for a.e. t ∈ R then dim E ≤ s + 1.
dim E = s + 1, where s is the essential supremum of dim(E ∩ (V + tv)) (t ∈ R).
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 3 / 14
Let n ≥ 2, E ⊂ Rn, v ∈ Sn−1 and V = v ⊥ ∈ G(n, n − 1).
If dim(E ∩ (V + tv)) ≤ s for a.e. t ∈ R then dim E ≤ s + 1.
dim E = s + 1, where s is the essential supremum of dim(E ∩ (V + tv)) (t ∈ R). How badly are these naive statements false?
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 3 / 14
Let n ≥ 2, E ⊂ Rn, v ∈ Sn−1 and V = v ⊥ ∈ G(n, n − 1).
If dim(E ∩ (V + tv)) ≤ s for a.e. t ∈ R then dim E ≤ s + 1.
dim E = s + 1, where s is the essential supremum of dim(E ∩ (V + tv)) (t ∈ R). How badly are these naive statements false? Very badly.
There exists a continuous function f : [0, 1] → [0, 1]n−1 such that dim(graph f ) = n. That is, for E = graph f we have dim E = n but s + 1 = 1 since dim(E ∩ (V + tv)) = 0 for every t.
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 3 / 14
Let n ≥ 2, E ⊂ Rn, v ∈ Sn−1 and V = v ⊥ ∈ G(n, n − 1).
dim E = s + 1, where s is the essential supremum of dim(E ∩ (V + tv)) (t ∈ R).
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 4 / 14
Let n ≥ 2, E ⊂ Rn, v ∈ Sn−1 and V = v ⊥ ∈ G(n, n − 1).
dim E = s + 1, where s is the essential supremum of dim(E ∩ (V + tv)) (t ∈ R).
We say that the set E has the Fubini property in direction v if the above naive Fubini holds.
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 4 / 14
Let n ≥ 2, E ⊂ Rn, v ∈ Sn−1 and V = v ⊥ ∈ G(n, n − 1).
dim E = s + 1, where s is the essential supremum of dim(E ∩ (V + tv)) (t ∈ R).
We say that the set E has the Fubini property in direction v if the above naive Fubini holds.
Every Borel set E ⊂ Rn has the Fubini property in almost every direction.
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 4 / 14
Let n ≥ 2, E ⊂ Rn, v ∈ Sn−1 and V = v ⊥ ∈ G(n, n − 1).
dim E = s + 1, where s is the essential supremum of dim(E ∩ (V + tv)) (t ∈ R).
We say that the set E has the Fubini property in direction v if the above naive Fubini holds.
Every Borel set E ⊂ Rn has the Fubini property in almost every direction. What can we say in a fixed direction?
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 4 / 14
Let n ≥ 2, E ⊂ Rn, v ∈ Sn−1 and V = v ⊥ ∈ G(n, n − 1).
We say that the set E has the Fubini property in direction v if dim E = s + 1, where s is the essential supremum of dim(E ∩ (V + tv)) (t ∈ R).
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 5 / 14
Let n ≥ 2, E ⊂ Rn, v ∈ Sn−1 and V = v ⊥ ∈ G(n, n − 1).
We say that the set E has the Fubini property in direction v if dim E = s + 1, where s is the essential supremum of dim(E ∩ (V + tv)) (t ∈ R).
A Borel set G ⊂ Rn is said to be Γ-null (in direction v) if for any Lipschitz curve γ with γ′, v > 0 a.e., the 1-dimensional Hausdorff measure of G ∩ γ is zero.
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 5 / 14
Let n ≥ 2, E ⊂ Rn, v ∈ Sn−1 and V = v ⊥ ∈ G(n, n − 1).
We say that the set E has the Fubini property in direction v if dim E = s + 1, where s is the essential supremum of dim(E ∩ (V + tv)) (t ∈ R).
A Borel set G ⊂ Rn is said to be Γ-null (in direction v) if for any Lipschitz curve γ with γ′, v > 0 a.e., the 1-dimensional Hausdorff measure of G ∩ γ is zero. Our main results says that by removing such a small set any Borel set can be made to have the Fubini property in any fixed direction:
For any v ∈ Sn−1 and Borel set E ⊂ Rn there is a Borel set G ⊂ E such that G is Γ-null in direction v and E \ G has the Fubini property in direction v.
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 5 / 14
Let n ≥ 2, E ⊂ Rn, v ∈ Sn−1 and V = v ⊥ ∈ G(n, n − 1).
We say that the set E has the Fubini property in direction v if dim E = s + 1, where s is the essential supremum of dim(E ∩ (V + tv)) (t ∈ R).
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 6 / 14
Let n ≥ 2, E ⊂ Rn, v ∈ Sn−1 and V = v ⊥ ∈ G(n, n − 1).
We say that the set E has the Fubini property in direction v if dim E = s + 1, where s is the essential supremum of dim(E ∩ (V + tv)) (t ∈ R). We pose the following conjecture:
Let E ⊂ Rn be a Borel set that can be obtained as a union of lines such that none
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 6 / 14
Let n ≥ 2, E ⊂ Rn, v ∈ Sn−1 and V = v ⊥ ∈ G(n, n − 1).
We say that the set E has the Fubini property in direction v if dim E = s + 1, where s is the essential supremum of dim(E ∩ (V + tv)) (t ∈ R). We pose the following conjecture:
Let E ⊂ Rn be a Borel set that can be obtained as a union of lines such that none
In the plane, using duality, projection theorem and Falconer-Mattila theorem about the dimension of union of lines, we can easily get:
The above conjecture holds for n = 2.
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 6 / 14
Let E ⊂ Rn be a Borel set that can be obtained as a union of lines such that none
A Borel set G ⊂ Rn is said to be Γ-null (in direction v) if for any Lipschitz curve γ with γ, v > 0 a.e. the 1-dimensional Hausdorff measure of G ∩ γ is zero.
For any v ∈ Sn−1 and Borel set E ⊂ Rn there is a Borel set G ⊂ E such that G is Γ-null in direction v and E \ G has the Fubini property in direction v.
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 7 / 14
Let E ⊂ Rn be a Borel set that can be obtained as a union of lines such that none
A Borel set G ⊂ Rn is said to be Γ-null (in direction v) if for any Lipschitz curve γ with γ, v > 0 a.e. the 1-dimensional Hausdorff measure of G ∩ γ is zero.
For any v ∈ Sn−1 and Borel set E ⊂ Rn there is a Borel set G ⊂ E such that G is Γ-null in direction v and E \ G has the Fubini property in direction v. Let E ⊂ Rn be a Borel set that can be obtained as a union of lines such that none
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 7 / 14
Let E ⊂ Rn be a Borel set that can be obtained as a union of lines such that none
A Borel set G ⊂ Rn is said to be Γ-null (in direction v) if for any Lipschitz curve γ with γ, v > 0 a.e. the 1-dimensional Hausdorff measure of G ∩ γ is zero.
For any v ∈ Sn−1 and Borel set E ⊂ Rn there is a Borel set G ⊂ E such that G is Γ-null in direction v and E \ G has the Fubini property in direction v. Let E ⊂ Rn be a Borel set that can be obtained as a union of lines such that none
By applying the Main Result to E we get a Borel set G ⊂ E such that G is Γ-null and E \ G has the Fubini property in direction v.
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 7 / 14
Let E ⊂ Rn be a Borel set that can be obtained as a union of lines such that none
A Borel set G ⊂ Rn is said to be Γ-null (in direction v) if for any Lipschitz curve γ with γ, v > 0 a.e. the 1-dimensional Hausdorff measure of G ∩ γ is zero.
For any v ∈ Sn−1 and Borel set E ⊂ Rn there is a Borel set G ⊂ E such that G is Γ-null in direction v and E \ G has the Fubini property in direction v. Let E ⊂ Rn be a Borel set that can be obtained as a union of lines such that none
By applying the Main Result to E we get a Borel set G ⊂ E such that G is Γ-null and E \ G has the Fubini property in direction v. Since G is Γ-null it intersects every line of E in a set of (1-dimensional Hausdorff measure) zero.
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 7 / 14
For any v ∈ Sn−1 and Borel set E ⊂ Rn there is a Borel set G ⊂ E such that G is Γ-null in direction v and E \ G has the Fubini property in direction v. Let E ⊂ Rn be a Borel set that can be obtained as a union of lines such that none
By applying the Main Result to E we get a Borel set G ⊂ E such that G is Γ-null and E \ G has the Fubini property in direction v. Since G is Γ-null it intersects every line of E in a set of (1-dimensional Hausdorff measure) zero.
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 8 / 14
For any v ∈ Sn−1 and Borel set E ⊂ Rn there is a Borel set G ⊂ E such that G is Γ-null in direction v and E \ G has the Fubini property in direction v. Let E ⊂ Rn be a Borel set that can be obtained as a union of lines such that none
By applying the Main Result to E we get a Borel set G ⊂ E such that G is Γ-null and E \ G has the Fubini property in direction v. Since G is Γ-null it intersects every line of E in a set of (1-dimensional Hausdorff measure) zero. Therefore we obtain the following:
Let E ⊂ Rn be a Borel set that can be obtained as a union of lines such that none
measure zero from each line we get a set with the Fubini property.
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 8 / 14
For any v ∈ Sn−1 and Borel set E ⊂ Rn there is a Borel set G ⊂ E such that G is Γ-null in direction v and E \ G has the Fubini property in direction v. Let E ⊂ Rn be a Borel set that can be obtained as a union of lines such that none
By applying the Main Result to E we get a Borel set G ⊂ E such that G is Γ-null and E \ G has the Fubini property in direction v. Since G is Γ-null it intersects every line of E in a set of (1-dimensional Hausdorff measure) zero. Therefore we obtain the following:
Let E ⊂ Rn be a Borel set that can be obtained as a union of lines such that none
measure zero from each line we get a set with the Fubini property. Can these sets of measure zero really make a difference?
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 8 / 14
Let E ⊂ Rn be a Borel set that can be obtained as a union of lines such that none
measure zero from each line we get a set with the Fubini property. Can these sets of measure zero really make a difference?
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 9 / 14
Let E ⊂ Rn be a Borel set that can be obtained as a union of lines such that none
measure zero from each line we get a set with the Fubini property. Can these sets of measure zero really make a difference? Recall the following from the previous talk:
Let L be a collection of lines in Rn, E be the union of the lines of L and let F ⊂ E be a set such that dim(F ∩ l) = 1 for every l ∈ L. Then dim F = dim E = dim L + 1 provided dim L ≤ 1 or dim E < 2.
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 9 / 14
Let E ⊂ Rn be a Borel set that can be obtained as a union of lines such that none
measure zero from each line we get a set with the Fubini property. Can these sets of measure zero really make a difference? Recall the following from the previous talk:
Let L be a collection of lines in Rn, E be the union of the lines of L and let F ⊂ E be a set such that dim(F ∩ l) = 1 for every l ∈ L. Then dim F = dim E = dim L + 1 provided dim L ≤ 1 or dim E < 2. Combining these we get the following partial result.
Let E ⊂ Rn be a Borel set that can be obtained as a union of lines such that none
property in direction v.
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 9 / 14
A Borel set B ⊂ Rn is a Besicovitch set if it contains a unit line segment in every direction. We say that a Borel set B ⊂ Rn is a line-Besicovitch set if it contains a line in every direction.
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 10 / 14
A Borel set B ⊂ Rn is a Besicovitch set if it contains a unit line segment in every direction. We say that a Borel set B ⊂ Rn is a line-Besicovitch set if it contains a line in every direction.
Every Besicovitch set in Rn has Hausdorff / Minkowski dimension n.
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 10 / 14
A Borel set B ⊂ Rn is a Besicovitch set if it contains a unit line segment in every direction. We say that a Borel set B ⊂ Rn is a line-Besicovitch set if it contains a line in every direction.
Every Besicovitch set in Rn has Hausdorff / Minkowski dimension n.
Let E ⊂ Rn be a Borel set that can be obtained as a union of lines such that none
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 10 / 14
A Borel set B ⊂ Rn is a Besicovitch set if it contains a unit line segment in every direction. We say that a Borel set B ⊂ Rn is a line-Besicovitch set if it contains a line in every direction.
Every Besicovitch set in Rn has Hausdorff / Minkowski dimension n.
Let E ⊂ Rn be a Borel set that can be obtained as a union of lines such that none
The above conjecture would imply that any line-Besicovitch set has Hausdorff dimension at least n − 1 and Minkowski dimension n.
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 10 / 14
If Fubini property holds for union of lines then any line-Besicovitch set has Hausdorff dimension at least n − 1.
It is easy to see that if we have a line-Besicovitch set with Hausdorff dimension d then we also have a Borel set B ⊂ Rn with dim B = d and with the property that for any point p of a fixed horizontal hyperplane H there exists a line lp, not far from vertical, such that lp \ {p} ⊂ B. Let C ⊂ H be the graph of a continuous function f : [0, 1] → [0, 1]n−2 with dim C = n − 1 and let E be the union of the lines lp for p ∈ C and let E ′ be the union of the punched lines lp \ {p} for p ∈ C. Then E ⊃ C, so dim E ≥ n − 1 but E ′ ⊂ B, so dim E ′ ≤ d. If v is chosen properly and V = v ⊥ then H ∩ (V + tv) is at most one point for every t, so dim E ′ ∩ (V + tv) = dim E ∩ (V + tv). By the always true Fubini inequality dim E ′ ∩ (V + tv) ≤ dim E ′ − 1 ≤ d − 1 for
dim E ∩ (V + tv) ≥ dim E − 1 ≥ n − 2 for a. e. t. Therefore d ≥ n − 1.
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 11 / 14
If Fubini property holds for union of lines then any line-Besicovitch set has Hausdorff dimension at least n − 1.
It is easy to see that if we have a line-Besicovitch set with Hausdorff dimension d then we also have a Borel set B ⊂ Rn with dim B = d and with the property that for any point p of a fixed horizontal hyperplane H there exists a line lp, not far from vertical, such that lp \ {p} ⊂ B. Let C ⊂ H be the graph of a continuous function f : [0, 1] → [0, 1]n−2 with dim C = n − 1 and let E be the union of the lines lp for p ∈ C and let E ′ be the union of the punched lines lp \ {p} for p ∈ C. Then E ⊃ C, so dim E ≥ n − 1 but E ′ ⊂ B, so dim E ′ ≤ d. If v is chosen properly and V = v ⊥ then H ∩ (V + tv) is at most one point for every t, so dim E ′ ∩ (V + tv) = dim E ∩ (V + tv). By the always true Fubini inequality dim E ′ ∩ (V + tv) ≤ dim E ′ − 1 ≤ d − 1 for
dim E ∩ (V + tv) ≥ dim E − 1 ≥ n − 2 for a. e. t. Therefore d ≥ n − 1.
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 11 / 14
If Fubini property holds for union of lines then any line-Besicovitch set has Hausdorff dimension at least n − 1.
It is easy to see that if we have a line-Besicovitch set with Hausdorff dimension d then we also have a Borel set B ⊂ Rn with dim B = d and with the property that for any point p of a fixed horizontal hyperplane H there exists a line lp, not far from vertical, such that lp \ {p} ⊂ B. Let C ⊂ H be the graph of a continuous function f : [0, 1] → [0, 1]n−2 with dim C = n − 1 and let E be the union of the lines lp for p ∈ C and let E ′ be the union of the punched lines lp \ {p} for p ∈ C. Then E ⊃ C, so dim E ≥ n − 1 but E ′ ⊂ B, so dim E ′ ≤ d. If v is chosen properly and V = v ⊥ then H ∩ (V + tv) is at most one point for every t, so dim E ′ ∩ (V + tv) = dim E ∩ (V + tv). By the always true Fubini inequality dim E ′ ∩ (V + tv) ≤ dim E ′ − 1 ≤ d − 1 for
dim E ∩ (V + tv) ≥ dim E − 1 ≥ n − 2 for a. e. t. Therefore d ≥ n − 1.
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 11 / 14
If Fubini property holds for union of lines then any line-Besicovitch set has Hausdorff dimension at least n − 1.
It is easy to see that if we have a line-Besicovitch set with Hausdorff dimension d then we also have a Borel set B ⊂ Rn with dim B = d and with the property that for any point p of a fixed horizontal hyperplane H there exists a line lp, not far from vertical, such that lp \ {p} ⊂ B. Let C ⊂ H be the graph of a continuous function f : [0, 1] → [0, 1]n−2 with dim C = n − 1 and let E be the union of the lines lp for p ∈ C and let E ′ be the union of the punched lines lp \ {p} for p ∈ C. Then E ⊃ C, so dim E ≥ n − 1 but E ′ ⊂ B, so dim E ′ ≤ d. If v is chosen properly and V = v ⊥ then H ∩ (V + tv) is at most one point for every t, so dim E ′ ∩ (V + tv) = dim E ∩ (V + tv). By the always true Fubini inequality dim E ′ ∩ (V + tv) ≤ dim E ′ − 1 ≤ d − 1 for
dim E ∩ (V + tv) ≥ dim E − 1 ≥ n − 2 for a. e. t. Therefore d ≥ n − 1.
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 11 / 14
If Fubini property holds for union of lines then any line-Besicovitch set has Hausdorff dimension at least n − 1.
It is easy to see that if we have a line-Besicovitch set with Hausdorff dimension d then we also have a Borel set B ⊂ Rn with dim B = d and with the property that for any point p of a fixed horizontal hyperplane H there exists a line lp, not far from vertical, such that lp \ {p} ⊂ B. Let C ⊂ H be the graph of a continuous function f : [0, 1] → [0, 1]n−2 with dim C = n − 1 and let E be the union of the lines lp for p ∈ C and let E ′ be the union of the punched lines lp \ {p} for p ∈ C. Then E ⊃ C, so dim E ≥ n − 1 but E ′ ⊂ B, so dim E ′ ≤ d. If v is chosen properly and V = v ⊥ then H ∩ (V + tv) is at most one point for every t, so dim E ′ ∩ (V + tv) = dim E ∩ (V + tv). By the always true Fubini inequality dim E ′ ∩ (V + tv) ≤ dim E ′ − 1 ≤ d − 1 for
dim E ∩ (V + tv) ≥ dim E − 1 ≥ n − 2 for a. e. t. Therefore d ≥ n − 1.
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 11 / 14
If Fubini property holds for union of lines then any line-Besicovitch set has Hausdorff dimension at least n − 1.
It is easy to see that if we have a line-Besicovitch set with Hausdorff dimension d then we also have a Borel set B ⊂ Rn with dim B = d and with the property that for any point p of a fixed horizontal hyperplane H there exists a line lp, not far from vertical, such that lp \ {p} ⊂ B. Let C ⊂ H be the graph of a continuous function f : [0, 1] → [0, 1]n−2 with dim C = n − 1 and let E be the union of the lines lp for p ∈ C and let E ′ be the union of the punched lines lp \ {p} for p ∈ C. Then E ⊃ C, so dim E ≥ n − 1 but E ′ ⊂ B, so dim E ′ ≤ d. If v is chosen properly and V = v ⊥ then H ∩ (V + tv) is at most one point for every t, so dim E ′ ∩ (V + tv) = dim E ∩ (V + tv). By the always true Fubini inequality dim E ′ ∩ (V + tv) ≤ dim E ′ − 1 ≤ d − 1 for
dim E ∩ (V + tv) ≥ dim E − 1 ≥ n − 2 for a. e. t. Therefore d ≥ n − 1.
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 11 / 14
Let m ∈ N and Pt : Rm × Rm → Rm (t ∈ R) be the following 1-dimensional family of projections: Pt(x, y) = tx + (1 − t)y.
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 12 / 14
Let m ∈ N and Pt : Rm × Rm → Rm (t ∈ R) be the following 1-dimensional family of projections: Pt(x, y) = tx + (1 − t)y.
For any Borel set B ⊂ R2m with dim B ≤ m, dim Pt(B) = dim B for
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 12 / 14
Let m ∈ N and Pt : Rm × Rm → Rm (t ∈ R) be the following 1-dimensional family of projections: Pt(x, y) = tx + (1 − t)y.
For any Borel set B ⊂ R2m with dim B ≤ m, dim Pt(B) = dim B for
For m = 1 this is essentially Martsrand’s classical projection theorem, and it gives the planar case of the Kakeya conjecture. For general m the same duality argument gives that this Conjecture would imply Kakeya conjecture in Rm+1. Combining the results of this talk (for n = m + 1) and the previous talk (for n = m + 1, k = 1) we get the following partial result about the above conjecture:
For any Borel set B ⊂ R2m with dim B ≤ 1, dim Pt(B) = dim B for a. e. t ∈ [0, 1].
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 12 / 14
Let m ∈ N and Pt : Rm × Rm → Rm (t ∈ R) be the following 1-dimensional family of projections: Pt(x, y) = tx + (1 − t)y. Combining the results of this talk (for n = m + 1) and the previous talk (for n = m + 1, k = 1) we get the following partial result about the above conjecture:
For any Borel set B ⊂ R2m with dim B ≤ 1, dim Pt(B) = dim B for a. e. t ∈ [0, 1].
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 13 / 14
Let m ∈ N and Pt : Rm × Rm → Rm (t ∈ R) be the following 1-dimensional family of projections: Pt(x, y) = tx + (1 − t)y. Combining the results of this talk (for n = m + 1) and the previous talk (for n = m + 1, k = 1) we get the following partial result about the above conjecture:
For any Borel set B ⊂ R2m with dim B ≤ 1, dim Pt(B) = dim B for a. e. t ∈ [0, 1].
If T = {t ∈ [0, 1] : dim Pt(B) ≤ dim B − ε} has positive measure and G is the Γ-null set such that ∪t∈TPt(B) \ G has the Fubini property then dim B + 1 = dim (∪t∈TPt(B) \ G) ≤ dim B − ε + 1.
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 13 / 14
Let m ∈ N and Pt : Rm × Rm → Rm (t ∈ R) be the following 1-dimensional family of projections: Pt(x, y) = tx + (1 − t)y. Combining the results of this talk (for n = m + 1) and the previous talk (for n = m + 1, k = 1) we get the following partial result about the above conjecture:
For any Borel set B ⊂ R2m with dim B ≤ 1, dim Pt(B) = dim B for a. e. t ∈ [0, 1].
If T = {t ∈ [0, 1] : dim Pt(B) ≤ dim B − ε} has positive measure and G is the Γ-null set such that ∪t∈TPt(B) \ G has the Fubini property then dim B + 1 = dim (∪t∈TPt(B) \ G) ≤ dim B − ε + 1. Sad news: This result can be also proved directly and quickly by using Kaufman’s potential theoretic method, see F¨ assler & Orponen.
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 13 / 14
Tam´ as Keleti (Budapest) with Korn´ elia H´ era and Andr´ as M´ ath´ e Warwick, 12 July 2017 14 / 14