Hausdorff dimension of union of affine subspaces Korn elia H era - PowerPoint PPT Presentation

Hausdorff dimension of union of affine subspaces Korn´ elia H´ era E¨ otv¨ os Lor´ and University, Budapest Workshop on Geometric Measure Theory University of Warwick July 10-14, 2017 Korn´ elia H´ era (E¨ otv¨ os University, Budapest) Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 1 / 19

Hausdorff dimension of union of affine subspaces Korn´ elia H´ era E¨ otv¨ os Lor´ and University, Budapest Workshop on Geometric Measure Theory University of Warwick July 10-14, 2017 joint work with Tam´ as Keleti and Andr´ as M´ ath´ e Korn´ elia H´ era (E¨ otv¨ os University, Budapest) Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 1 / 19

The heuristic principle and an example The union of an s -Hausdorff-dimensional collection of d -dimensional sets in R n must have Hausdorff dimension s + d if s + d ≤ n and positive measure if s + d > n . Korn´ elia H´ era (E¨ otv¨ os University, Budapest) Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 2 / 19

The heuristic principle and an example The union of an s -Hausdorff-dimensional collection of d -dimensional sets in R n must have Hausdorff dimension s + d if s + d ≤ n and positive measure if s + d > n . In this talk dimension always refers to Hausdorff dimension. Korn´ elia H´ era (E¨ otv¨ os University, Budapest) Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 2 / 19

The heuristic principle and an example The union of an s -Hausdorff-dimensional collection of d -dimensional sets in R n must have Hausdorff dimension s + d if s + d ≤ n and positive measure if s + d > n . In this talk dimension always refers to Hausdorff dimension. Results for collections of spheres By the union of an s -dimensional collection of spheres in R n we mean a union ( x , r ) ∈ E ( x + rS n − 1 ) ⊂ R n , where ∅ � = E ⊂ R n × (0 , ∞ ) with dim E = s . � Korn´ elia H´ era (E¨ otv¨ os University, Budapest) Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 2 / 19

The heuristic principle and an example The union of an s -Hausdorff-dimensional collection of d -dimensional sets in R n must have Hausdorff dimension s + d if s + d ≤ n and positive measure if s + d > n . In this talk dimension always refers to Hausdorff dimension. Results for collections of spheres By the union of an s -dimensional collection of spheres in R n we mean a union ( x , r ) ∈ E ( x + rS n − 1 ) ⊂ R n , where ∅ � = E ⊂ R n × (0 , ∞ ) with dim E = s . � If s ≤ 1 then the union of any s -dimensional collection of spheres in R n ( n ≥ 2) has dimension s + n − 1 (Wolff (2000), Oberlin (2006)). Korn´ elia H´ era (E¨ otv¨ os University, Budapest) Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 2 / 19

The heuristic principle and an example The union of an s -Hausdorff-dimensional collection of d -dimensional sets in R n must have Hausdorff dimension s + d if s + d ≤ n and positive measure if s + d > n . In this talk dimension always refers to Hausdorff dimension. Results for collections of spheres By the union of an s -dimensional collection of spheres in R n we mean a union ( x , r ) ∈ E ( x + rS n − 1 ) ⊂ R n , where ∅ � = E ⊂ R n × (0 , ∞ ) with dim E = s . � If s ≤ 1 then the union of any s -dimensional collection of spheres in R n ( n ≥ 2) has dimension s + n − 1 (Wolff (2000), Oberlin (2006)). If s > 1 then the union of any s -dimensional collection of spheres in R n ( n ≥ 2) has positive Lebesgue measure (Mitsis (1999) , Wolff (2000), Oberlin (2006)). Korn´ elia H´ era (E¨ otv¨ os University, Budapest) Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 2 / 19

The heuristic principle and an example The union of an s -Hausdorff-dimensional collection of d -dimensional sets in R n must have Hausdorff dimension s + d if s + d ≤ n and positive measure if s + d > n . In this talk dimension always refers to Hausdorff dimension. Results for collections of spheres By the union of an s -dimensional collection of spheres in R n we mean a union ( x , r ) ∈ E ( x + rS n − 1 ) ⊂ R n , where ∅ � = E ⊂ R n × (0 , ∞ ) with dim E = s . � If s ≤ 1 then the union of any s -dimensional collection of spheres in R n ( n ≥ 2) has dimension s + n − 1 (Wolff (2000), Oberlin (2006)). If s > 1 then the union of any s -dimensional collection of spheres in R n ( n ≥ 2) has positive Lebesgue measure (Mitsis (1999) , Wolff (2000), Oberlin (2006)). ( x , r ) ∈ E ( x + rS n − 1 ) can be The upper bound s + n − 1 in the first case is easy: � obtained as a Lipschitz image of E × S n − 1 . Korn´ elia H´ era (E¨ otv¨ os University, Budapest) Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 2 / 19

The heuristic principle and an example The union of an s -Hausdorff-dimensional collection of d -dimensional sets in R n must have Hausdorff dimension s + d if s + d ≤ n and positive measure if s + d > n . In this talk dimension always refers to Hausdorff dimension. Results for collections of spheres By the union of an s -dimensional collection of spheres in R n we mean a union ( x , r ) ∈ E ( x + rS n − 1 ) ⊂ R n , where ∅ � = E ⊂ R n × (0 , ∞ ) with dim E = s . � If s ≤ 1 then the union of any s -dimensional collection of spheres in R n ( n ≥ 2) has dimension s + n − 1 (Wolff (2000), Oberlin (2006)). If s > 1 then the union of any s -dimensional collection of spheres in R n ( n ≥ 2) has positive Lebesgue measure (Mitsis (1999) , Wolff (2000), Oberlin (2006)). ( x , r ) ∈ E ( x + rS n − 1 ) can be The upper bound s + n − 1 in the first case is easy: � obtained as a Lipschitz image of E × S n − 1 . Thus the heuristic principle holds for collections of spheres. Korn´ elia H´ era (E¨ otv¨ os University, Budapest) Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 2 / 19

Affine subspaces Results for hyperplanes (Oberlin (2006), Falconer and Mattila (2016)) The union of any nonempty s -dimensional family of affine hyperplanes in R n has dimension s + n − 1 if s ∈ [0 , 1], and positive Lebesgue-measure if s > 1 . Thus the heuristic principle holds for collections of hyperplanes. Korn´ elia H´ era (E¨ otv¨ os University, Budapest) Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 3 / 19

Affine subspaces Results for hyperplanes (Oberlin (2006), Falconer and Mattila (2016)) The union of any nonempty s -dimensional family of affine hyperplanes in R n has dimension s + n − 1 if s ∈ [0 , 1], and positive Lebesgue-measure if s > 1 . Thus the heuristic principle holds for collections of hyperplanes. Moreover, Falconer and Mattila proved that instead of full hyperplanes, it is enough to take a positive H n − 1 -measure subset of each of them. Korn´ elia H´ era (E¨ otv¨ os University, Budapest) Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 3 / 19

Affine subspaces Results for hyperplanes (Oberlin (2006), Falconer and Mattila (2016)) The union of any nonempty s -dimensional family of affine hyperplanes in R n has dimension s + n − 1 if s ∈ [0 , 1], and positive Lebesgue-measure if s > 1 . Thus the heuristic principle holds for collections of hyperplanes. Moreover, Falconer and Mattila proved that instead of full hyperplanes, it is enough to take a positive H n − 1 -measure subset of each of them. More generally: for k -dimensional affine subspaces (1 ≤ k ≤ n − 1)? Korn´ elia H´ era (E¨ otv¨ os University, Budapest) Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 3 / 19

Affine subspaces Results for hyperplanes (Oberlin (2006), Falconer and Mattila (2016)) The union of any nonempty s -dimensional family of affine hyperplanes in R n has dimension s + n − 1 if s ∈ [0 , 1], and positive Lebesgue-measure if s > 1 . Thus the heuristic principle holds for collections of hyperplanes. Moreover, Falconer and Mattila proved that instead of full hyperplanes, it is enough to take a positive H n − 1 -measure subset of each of them. More generally: for k -dimensional affine subspaces (1 ≤ k ≤ n − 1)? Let A ( n , k ) denote the space of all k -dimensional affine subspaces of R n and consider any natural metric on A ( n , k ). By an s -dimensional family of k -dimensional affine subspaces we mean a set ∅ � = E ⊂ A ( n , k ) with dim E = s . Korn´ elia H´ era (E¨ otv¨ os University, Budapest) Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 3 / 19

Affine subspaces Theorem (H-Keleti-M´ ath´ e) For any integers 1 ≤ k < n and s ∈ [0 , 1], the union of any nonempty s -Hausdorff-dimensional family of k -dimensional affine subspaces of R n has Hausdorff dimension s + k . Korn´ elia H´ era (E¨ otv¨ os University, Budapest) Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 4 / 19