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 and Andr´ as M´ ath´ e Warwick, 12 July 2017 1 / 14

True and false Fubini inequalities for Hausdorff dimension Let n ≥ 2, E ⊂ R n , v ∈ S n − 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

True and false Fubini inequalities for Hausdorff dimension Let n ≥ 2, E ⊂ R n , v ∈ S n − 1 and V = v ⊥ ∈ G ( n , n − 1). Folklore (?) Fubini type inequality for Hausdorff dimension 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

True and false Fubini inequalities for Hausdorff dimension Let n ≥ 2, E ⊂ R n , v ∈ S n − 1 and V = v ⊥ ∈ G ( n , n − 1). Folklore (?) Fubini type inequality for Hausdorff dimension If { t ∈ R : dim( E ∩ ( V + tv )) ≥ s } has positive Lebesgue measure then dim E ≥ s + 1. (In general false) naive reverse inequality 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

True and false Fubini inequalities for Hausdorff dimension Let n ≥ 2, E ⊂ R n , v ∈ S n − 1 and V = v ⊥ ∈ G ( n , n − 1). Folklore (?) Fubini type inequality for Hausdorff dimension If { t ∈ R : dim( E ∩ ( V + tv )) ≥ s } has positive Lebesgue measure then dim E ≥ s + 1. (In general false) naive reverse inequality If dim( E ∩ ( V + tv )) ≤ s for a.e. t ∈ R then dim E ≤ s + 1. Combining the above inequalities we would get: (In general false) naive Fubini 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

Fubini is false for Hausdorff dimension Let n ≥ 2, E ⊂ R n , v ∈ S n − 1 and V = v ⊥ ∈ G ( n , n − 1). (In general false) naive reverse inequality If dim( E ∩ ( V + tv )) ≤ s for a.e. t ∈ R then dim E ≤ s + 1. (In general false) naive Fubini 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

Fubini is false for Hausdorff dimension Let n ≥ 2, E ⊂ R n , v ∈ S n − 1 and V = v ⊥ ∈ G ( n , n − 1). (In general false) naive reverse inequality If dim( E ∩ ( V + tv )) ≤ s for a.e. t ∈ R then dim E ≤ s + 1. (In general false) naive Fubini 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

Fubini is false for Hausdorff dimension Let n ≥ 2, E ⊂ R n , v ∈ S n − 1 and V = v ⊥ ∈ G ( n , n − 1). (In general false) naive reverse inequality If dim( E ∩ ( V + tv )) ≤ s for a.e. t ∈ R then dim E ≤ s + 1. (In general false) naive Fubini 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. Folklore 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

Marstrand-Mattila theorem Let n ≥ 2, E ⊂ R n , v ∈ S n − 1 and V = v ⊥ ∈ G ( n , n − 1). (In general false) naive Fubini 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

Marstrand-Mattila theorem Let n ≥ 2, E ⊂ R n , v ∈ S n − 1 and V = v ⊥ ∈ G ( n , n − 1). (In general false) naive Fubini dim E = s + 1, where s is the essential supremum of dim( E ∩ ( V + tv )) ( t ∈ R ). Definition 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

Marstrand-Mattila theorem Let n ≥ 2, E ⊂ R n , v ∈ S n − 1 and V = v ⊥ ∈ G ( n , n − 1). (In general false) naive Fubini dim E = s + 1, where s is the essential supremum of dim( E ∩ ( V + tv )) ( t ∈ R ). Definition We say that the set E has the Fubini property in direction v if the above naive Fubini holds. Theorem (Marstrand for n = 2, Mattila for n > 2) Every Borel set E ⊂ R n 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

Marstrand-Mattila theorem Let n ≥ 2, E ⊂ R n , v ∈ S n − 1 and V = v ⊥ ∈ G ( n , n − 1). (In general false) naive Fubini dim E = s + 1, where s is the essential supremum of dim( E ∩ ( V + tv )) ( t ∈ R ). Definition We say that the set E has the Fubini property in direction v if the above naive Fubini holds. Theorem (Marstrand for n = 2, Mattila for n > 2) Every Borel set E ⊂ R n 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

Our main result: a weaker Fubini in a fixed direction Let n ≥ 2, E ⊂ R n , v ∈ S n − 1 and V = v ⊥ ∈ G ( n , n − 1). Definition 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

Our main result: a weaker Fubini in a fixed direction Let n ≥ 2, E ⊂ R n , v ∈ S n − 1 and V = v ⊥ ∈ G ( n , n − 1). Definition 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 ). Definition (Alberti - Cs¨ ornyei - Preiss) A Borel set G ⊂ R n 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

Our main result: a weaker Fubini in a fixed direction Let n ≥ 2, E ⊂ R n , v ∈ S n − 1 and V = v ⊥ ∈ G ( n , n − 1). Definition 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 ). Definition (Alberti - Cs¨ ornyei - Preiss) A Borel set G ⊂ R n 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: Main Result For any v ∈ S n − 1 and Borel set E ⊂ R n 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

Fubini property of union of lines Let n ≥ 2, E ⊂ R n , v ∈ S n − 1 and V = v ⊥ ∈ G ( n , n − 1). Definition 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

Fubini property of union of lines Let n ≥ 2, E ⊂ R n , v ∈ S n − 1 and V = v ⊥ ∈ G ( n , n − 1). Definition 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: Conjecture Let E ⊂ R n be a Borel set that can be obtained as a union of lines such that none of the lines is orthogonal to v . Then E 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 6 / 14

Fubini property of union of lines Let n ≥ 2, E ⊂ R n , v ∈ S n − 1 and V = v ⊥ ∈ G ( n , n − 1). Definition 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: Conjecture Let E ⊂ R n be a Borel set that can be obtained as a union of lines such that none of the lines is orthogonal to v . Then E has the Fubini property in direction v . In the plane, using duality, projection theorem and Falconer-Mattila theorem about the dimension of union of lines, we can easily get: Theorem 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