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

SLIDE 1

Hausdorff dimension of union of affine subspaces

Korn´ elia H´ era

E¨

tv¨
s Lor´

and University, Budapest

Workshop on Geometric Measure Theory University of Warwick July 10-14, 2017

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 1 / 19

SLIDE 2

Hausdorff dimension of union of affine subspaces

Korn´ elia H´ era

E¨

tv¨
s 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¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 1 / 19

SLIDE 3

The heuristic principle and an example

The union of an s-Hausdorff-dimensional collection of d-dimensional sets in Rn must have Hausdorff dimension s + d if s + d ≤ n and positive measure if s + d > n.

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 2 / 19

SLIDE 4

The heuristic principle and an example

The union of an s-Hausdorff-dimensional collection of d-dimensional sets in Rn 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¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 2 / 19

SLIDE 5

The heuristic principle and an example

The union of an s-Hausdorff-dimensional collection of d-dimensional sets in Rn 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 Rn we mean a union

(x,r)∈E(x + rSn−1) ⊂ Rn, where ∅ = E ⊂ Rn × (0, ∞) with dim E = s.

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 2 / 19

SLIDE 6

The heuristic principle and an example

The union of an s-Hausdorff-dimensional collection of d-dimensional sets in Rn 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 Rn we mean a union

(x,r)∈E(x + rSn−1) ⊂ Rn, where ∅ = E ⊂ Rn × (0, ∞) with dim E = s.

If s ≤ 1 then the union of any s-dimensional collection of spheres in Rn (n ≥ 2) has dimension s + n − 1 (Wolff (2000), Oberlin (2006)).

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 2 / 19

SLIDE 7

The heuristic principle and an example

The union of an s-Hausdorff-dimensional collection of d-dimensional sets in Rn 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 Rn we mean a union

(x,r)∈E(x + rSn−1) ⊂ Rn, where ∅ = E ⊂ Rn × (0, ∞) with dim E = s.

If s ≤ 1 then the union of any s-dimensional collection of spheres in Rn (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 Rn (n ≥ 2) has positive Lebesgue measure (Mitsis (1999) , Wolff (2000), Oberlin (2006)).

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 2 / 19

SLIDE 8

The heuristic principle and an example

The union of an s-Hausdorff-dimensional collection of d-dimensional sets in Rn 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 Rn we mean a union

(x,r)∈E(x + rSn−1) ⊂ Rn, where ∅ = E ⊂ Rn × (0, ∞) with dim E = s.

If s ≤ 1 then the union of any s-dimensional collection of spheres in Rn (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 Rn (n ≥ 2) has positive Lebesgue measure (Mitsis (1999) , Wolff (2000), Oberlin (2006)). The upper bound s + n − 1 in the first case is easy:

(x,r)∈E(x + rSn−1) can be

btained as a Lipschitz image of E × Sn−1.

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 2 / 19

SLIDE 9

The heuristic principle and an example

The union of an s-Hausdorff-dimensional collection of d-dimensional sets in Rn 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 Rn we mean a union

(x,r)∈E(x + rSn−1) ⊂ Rn, where ∅ = E ⊂ Rn × (0, ∞) with dim E = s.

If s ≤ 1 then the union of any s-dimensional collection of spheres in Rn (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 Rn (n ≥ 2) has positive Lebesgue measure (Mitsis (1999) , Wolff (2000), Oberlin (2006)). The upper bound s + n − 1 in the first case is easy:

(x,r)∈E(x + rSn−1) can be

btained as a Lipschitz image of E × Sn−1.

Thus the heuristic principle holds for collections of spheres.

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 2 / 19

SLIDE 10

Affine subspaces

Results for hyperplanes (Oberlin (2006), Falconer and Mattila (2016))

The union of any nonempty s-dimensional family of affine hyperplanes in Rn 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¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 3 / 19

SLIDE 11

Affine subspaces

Results for hyperplanes (Oberlin (2006), Falconer and Mattila (2016))

The union of any nonempty s-dimensional family of affine hyperplanes in Rn 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 Hn−1-measure subset of each of them.

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 3 / 19

SLIDE 12

Affine subspaces

Results for hyperplanes (Oberlin (2006), Falconer and Mattila (2016))

The union of any nonempty s-dimensional family of affine hyperplanes in Rn 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 Hn−1-measure subset of each of them. More generally: for k-dimensional affine subspaces (1 ≤ k ≤ n − 1)?

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 3 / 19

SLIDE 13

Affine subspaces

Results for hyperplanes (Oberlin (2006), Falconer and Mattila (2016))

The union of any nonempty s-dimensional family of affine hyperplanes in Rn 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 Hn−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 Rn 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¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 3 / 19

SLIDE 14

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 Rn has Hausdorff dimension s + k.

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 4 / 19

SLIDE 15

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 Rn has Hausdorff dimension s + k. The upper bound is easy: For any 1 ≤ k < n integers and ∅ = E ⊂ A(n, k) we have dim (

P∈E P) ≤ k + dim E.

Proof: E can be decomposed into finitely many parts Ei such that for each i,

P∈Ei P is obtained as a Lipschitz image of Ei × Rk.

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 4 / 19

SLIDE 16

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 Rn has Hausdorff dimension s + k. The upper bound is easy: For any 1 ≤ k < n integers and ∅ = E ⊂ A(n, k) we have dim (

P∈E P) ≤ k + dim E.

Proof: E can be decomposed into finitely many parts Ei such that for each i,

P∈Ei P is obtained as a Lipschitz image of Ei × Rk.

The lower bound is harder.

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 4 / 19

SLIDE 17

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 Rn has Hausdorff dimension s + k. The upper bound is easy: For any 1 ≤ k < n integers and ∅ = E ⊂ A(n, k) we have dim (

P∈E P) ≤ k + dim E.

Proof: E can be decomposed into finitely many parts Ei such that for each i,

P∈Ei P is obtained as a Lipschitz image of Ei × Rk.

The lower bound is harder. Thus the heuristic principle holds for at most 1-dimensional collections of affine subspaces.

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 4 / 19

SLIDE 18

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 Rn has Hausdorff dimension s + k. The upper bound is easy: For any 1 ≤ k < n integers and ∅ = E ⊂ A(n, k) we have dim (

P∈E P) ≤ k + dim E.

Proof: E can be decomposed into finitely many parts Ei such that for each i,

P∈Ei P is obtained as a Lipschitz image of Ei × Rk.

The lower bound is harder. Thus the heuristic principle holds for at most 1-dimensional collections of affine subspaces. On the other hand, the heuristic principle is not true generally if s > 1: For any collection of lines of a fixed plane of R3 the union clearly has Hausdorff dimension at most 2, which is less than s + 1 if s > 1.

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 4 / 19

SLIDE 19

Affine subspaces

Theorem

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 Rn has Hausdorff dimension s + k. We can go further: it is enough to take a k-dimensional subset of each of the affine subspaces:

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 5 / 19

SLIDE 20

Affine subspaces

Theorem

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 Rn has Hausdorff dimension s + k. We can go further: it is enough to take a k-dimensional subset of each of the affine subspaces:

Theorem (H-Keleti-M´ ath´ e)

Let 1 ≤ k < n be integers and s ∈ [0, 1]. If E is a nonempty s-Hausdorff dimensional family of k-dimensional affine subspaces and B ⊂ Rn such that dim (B ∩ P) = k for every P ∈ E then dim B ≥ s + k. if B ⊂

P∈E P such that dim (B ∩ P) = k for every P ∈ E then

dim B = dim

P∈E P
= s + k.

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 5 / 19

SLIDE 21

Affine subspaces

Theorem

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 Rn has Hausdorff dimension s + k. We can go further: it is enough to take a k-dimensional subset of each of the affine subspaces:

Theorem (H-Keleti-M´ ath´ e)

Let 1 ≤ k < n be integers and s ∈ [0, 1]. If E is a nonempty s-Hausdorff dimensional family of k-dimensional affine subspaces and B ⊂ Rn such that dim (B ∩ P) = k for every P ∈ E then dim B ≥ s + k. if B ⊂

P∈E P such that dim (B ∩ P) = k for every P ∈ E then

dim B = dim

P∈E P
= s + k.

If the equality on the left hand side holds for k = 1 in Rn for all n ≥ 2, it would imply that Besicovitch sets in Rn have Hausdorff dimension at least n − 1 and upper Minkowski dimension n (see Keleti, 2016).

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 5 / 19

SLIDE 22

Results for cube skeletons

Recall from previous talk:

Theorem (Chang-Cs¨

rnyei-H-Keleti)

For any integers 0 ≤ k < n, the minimal Hausdorff dimension of a Borel set A ⊂ Rn that contains the k-skeleton of

1

a scaled copy of a cube around every point of Rn is n − 1;

2

a scaled and rotated copy of a cube around every point of Rn is k;

3

a rotated copy of a cube around every point of Rn is k + 1;

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 6 / 19

SLIDE 23

Results for cube skeletons

Recall from previous talk:

Theorem (Chang-Cs¨

rnyei-H-Keleti)

For any integers 0 ≤ k < n, the minimal Hausdorff dimension of a Borel set A ⊂ Rn that contains the k-skeleton of

1

a scaled copy of a cube around every point of Rn is n − 1;

2

a scaled and rotated copy of a cube around every point of Rn is k;

3

a rotated copy of a cube around every point of Rn is k + 1;

Lower estimates

1

The lower bound n − 1 in the first case is easy.

2

The lower bound k in the second case is trivial.

3

The lower bound k + 1 in the third case is nontrivial.

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 6 / 19

SLIDE 24

Results for cube skeletons

Theorem (H-Keleti-M´ ath´ e)

If B ⊂ Rn contains the k-skeleton of a rotated unit cube around every point of Rn, then dim B ≥ k + 1.

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 7 / 19

SLIDE 25

Results for cube skeletons

Theorem (H-Keleti-M´ ath´ e)

If B ⊂ Rn contains the k-skeleton of a rotated unit cube around every point of Rn, then dim B ≥ k + 1. Idea of proof:

Theorem

Let 1 ≤ k < n be integers and s ∈ [0, 1]. If E is a nonempty s-Hausdorff dimensional family of k-dimensional affine subspaces and B ⊂ Rn such that dim (B ∩ P) = k for every P ∈ E then dim B ≥ s + k. Define E ⊂ A(n, k) as the set of the k-dimensional affine subspaces of the k-faces

f the cubes contained in B. It can be shown that the condition above implies

dim E ≥ 1, thus by our theorem for 1-dimensional collections of affine subspaces, dim B ≥ k + 1.

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 7 / 19

SLIDE 26

Main Result

What happens if instead of dim (B ∩ P) = k we have dim (B ∩ P) ≥ α for all P ∈ E?

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 8 / 19

SLIDE 27

Main Result

What happens if instead of dim (B ∩ P) = k we have dim (B ∩ P) ≥ α for all P ∈ E?

Theorem (H-Keleti-M´ ath´ e)

Let 1 ≤ k < n be integers, and 0 < α ≤ k be any real number. Suppose that B ⊂ Rn, ∅ = E ⊂ A(n, k) and for every k-dimensional affine subspace P ∈ E, dim (P ∩ B) ≥ α. Then dim B ≥ 2α − k + min(dim E, 1).

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 8 / 19

SLIDE 28

Main Result

What happens if instead of dim (B ∩ P) = k we have dim (B ∩ P) ≥ α for all P ∈ E?

Theorem (H-Keleti-M´ ath´ e)

Let 1 ≤ k < n be integers, and 0 < α ≤ k be any real number. Suppose that B ⊂ Rn, ∅ = E ⊂ A(n, k) and for every k-dimensional affine subspace P ∈ E, dim (P ∩ B) ≥ α. Then dim B ≥ 2α − k + min(dim E, 1). This can be seen as a lower estimate for the Hausdorff dimension of generalized Furstenberg-type sets.

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 8 / 19

SLIDE 29

Main Result

What happens if instead of dim (B ∩ P) = k we have dim (B ∩ P) ≥ α for all P ∈ E?

Theorem (H-Keleti-M´ ath´ e)

Let 1 ≤ k < n be integers, and 0 < α ≤ k be any real number. Suppose that B ⊂ Rn, ∅ = E ⊂ A(n, k) and for every k-dimensional affine subspace P ∈ E, dim (P ∩ B) ≥ α. Then dim B ≥ 2α − k + min(dim E, 1). This can be seen as a lower estimate for the Hausdorff dimension of generalized Furstenberg-type sets.

Classical Furstenberg sets

Let 0 < α ≤ 1. A set F ⊂ R2 is a Furstenberg set, if for every e ∈ S1 there is a line Le in direction e such that dim Le ∩ F ≥ α. If F is a Furstenberg-set, then dim F ≥ 2α, dim F ≥ α + 1

2 (Wolff, 2003).

There exists a Furstenberg set with dim F = 3α

2 + 1 2 (Wolff, 2003).

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 8 / 19

SLIDE 30

Furstenberg-type sets

Classical Furstenberg sets

Let 0 < α ≤ 1. A set F ⊂ R2 is a Furstenberg set, if for every e ∈ S1 there is a line Le in direction e such that dim Le ∩ F ≥ α. If F is a Furstenberg-set, then dim F ≥ 2α, dim F ≥ α + 1

2 (Wolff, 2003).

There exists a Furstenberg set with dim F = 3α

2 + 1 2 (Wolff, 2003).

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 9 / 19

SLIDE 31

Furstenberg-type sets

Classical Furstenberg sets

Let 0 < α ≤ 1. A set F ⊂ R2 is a Furstenberg set, if for every e ∈ S1 there is a line Le in direction e such that dim Le ∩ F ≥ α. If F is a Furstenberg-set, then dim F ≥ 2α, dim F ≥ α + 1

2 (Wolff, 2003).

There exists a Furstenberg set with dim F = 3α

2 + 1 2 (Wolff, 2003).

In our theorem B is a generalized Furstenberg-type set: it intersects every element of a given family of k-dimensional affine subspaces in a set of Hausdorff dimension at least α.

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 9 / 19

SLIDE 32

Furstenberg-type sets

Classical Furstenberg sets

Let 0 < α ≤ 1. A set F ⊂ R2 is a Furstenberg set, if for every e ∈ S1 there is a line Le in direction e such that dim Le ∩ F ≥ α. If F is a Furstenberg-set, then dim F ≥ 2α, dim F ≥ α + 1

2 (Wolff, 2003).

There exists a Furstenberg set with dim F = 3α

2 + 1 2 (Wolff, 2003).

In our theorem B is a generalized Furstenberg-type set: it intersects every element of a given family of k-dimensional affine subspaces in a set of Hausdorff dimension at least α. Our theorem is a generalization of the first Furstenberg estimate: It gives that if F ⊂ Rn is a Furstenberg set, then the lines Le form an at least 1-dimensional collection of lines of R2 and thus dim F ≥ 2α − 1 + 1 = 2α.

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 9 / 19

SLIDE 33

Outline of the proof of our main result

Theorem (H-Keleti-M´ ath´ e)

Let 1 ≤ k < n be integers, and 0 < α ≤ k be any real number. Suppose that B ⊂ Rn, ∅ = E ⊂ A(n, k) and for every k-dimensional affine subspace P ∈ E, dim (P ∩ B) ≥ α. Then dim B ≥ 2α − k + min(dim E, 1).

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 10 / 19

SLIDE 34

Outline of the proof of our main result

Theorem (H-Keleti-M´ ath´ e)

Let 1 ≤ k < n be integers, and 0 < α ≤ k be any real number. Suppose that B ⊂ Rn, ∅ = E ⊂ A(n, k) and for every k-dimensional affine subspace P ∈ E, dim (P ∩ B) ≥ α. Then dim B ≥ 2α − k + min(dim E, 1). Main ingredients of the proof: An L2-estimate, which produces a lower bound for the Lebesgue-measure of the δ-neighborhood of B.

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 10 / 19

SLIDE 35

Outline of the proof of our main result

Theorem (H-Keleti-M´ ath´ e)

Let 1 ≤ k < n be integers, and 0 < α ≤ k be any real number. Suppose that B ⊂ Rn, ∅ = E ⊂ A(n, k) and for every k-dimensional affine subspace P ∈ E, dim (P ∩ B) ≥ α. Then dim B ≥ 2α − k + min(dim E, 1). Main ingredients of the proof: An L2-estimate, which produces a lower bound for the Lebesgue-measure of the δ-neighborhood of B. A technical argument, which makes it possible to estimate the Hausdorff dimension of B instead of the Minkowski dimension.

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 10 / 19

SLIDE 36

Outline of the proof of our main result

Theorem (H-Keleti-M´ ath´ e)

Let 1 ≤ k < n be integers, and 0 < α ≤ k be any real number. Suppose that B ⊂ Rn, ∅ = E ⊂ A(n, k) and for every k-dimensional affine subspace P ∈ E, dim (P ∩ B) ≥ α. Then dim B ≥ 2α − k + min(dim E, 1). Main ingredients of the proof: An L2-estimate, which produces a lower bound for the Lebesgue-measure of the δ-neighborhood of B. A technical argument, which makes it possible to estimate the Hausdorff dimension of B instead of the Minkowski dimension. Geometric considerations about the intersection of pairs of δ-tubes: Fix a bounded set S (e.g. a cube), and cut each affine subspace P with S. Then the δ-neighborhood of P ∩ S is called a δ-tube. The Lebesgue measure of a δ-tube is ≈ δn−k.

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 10 / 19

SLIDE 37

L2-estimate

We fix a Frostman measure µ on E ⊂ A(n, k).

The Cauchy-Schwarz inequality

E

Ln((P ∩ B)δ)dµ(P) ≤

Rn

χBδ(y) ·

E

χPδ(y)dµ(P)dy ≤ ≤  

Rn

χ2

Bδ(y)dy

 

1/2

·   

Rn

 

E

χPδ(y)dµ(P)  

2

dy   

1/2

= = (Ln(Bδ))1/2 ·  

E×E

Ln (Pδ ∩ P′

δ) dµ(P)dµ(P′)

 

1/2

.

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 11 / 19

SLIDE 38

L2-estimate

We fix a Frostman measure µ on E ⊂ A(n, k).

The Cauchy-Schwarz inequality

E

Ln((P ∩ B)δ)dµ(P) ≤

Rn

χBδ(y) ·

E

χPδ(y)dµ(P)dy ≤ ≤  

Rn

χ2

Bδ(y)dy

 

1/2

·   

Rn

 

E

χPδ(y)dµ(P)  

2

dy   

1/2

= = (Ln(Bδ))1/2 ·  

E×E

Ln (Pδ ∩ P′

δ) dµ(P)dµ(P′)

 

1/2

. We want to estimate Ln(Bδ) from below.

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 11 / 19

SLIDE 39

L2-estimate

We fix a Frostman measure µ on E ⊂ A(n, k).

The Cauchy-Schwarz inequality

E

Ln((P ∩ B)δ)dµ(P) ≤

Rn

χBδ(y) ·

E

χPδ(y)dµ(P)dy ≤ ≤  

Rn

χ2

Bδ(y)dy

 

1/2

·   

Rn

 

E

χPδ(y)dµ(P)  

2

dy   

1/2

= = (Ln(Bδ))1/2 ·  

E×E

Ln (Pδ ∩ P′

δ) dµ(P)dµ(P′)

 

1/2

. We want to estimate Ln(Bδ) from below. Left hand side: If Hα

∞(P ∩ B) ≥ ε for all P ∈ E (this can be assumed without loss of generality),

then

E

Ln((P ∩ B)δ)dµ(P) δn−α.

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 11 / 19

SLIDE 40

L2-estimate and geometric arguments

The Cauchy-Schwarz inequality

E

Ln((P ∩ B)δ)dµ(P) ≤ (Ln(Bδ))1/2 ·

E×E

Ln (Pδ ∩ P′

δ) dµ(P)dµ(P′)

1/2 . We need an upper estimate for

E×E

Ln (Pδ ∩ P′

δ) dµ(P)dµ(P′).

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 12 / 19

SLIDE 41

L2-estimate and geometric arguments

The Cauchy-Schwarz inequality

E

Ln((P ∩ B)δ)dµ(P) ≤ (Ln(Bδ))1/2 ·

E×E

Ln (Pδ ∩ P′

δ) dµ(P)dµ(P′)

1/2 . We need an upper estimate for

E×E

Ln (Pδ ∩ P′

δ) dµ(P)dµ(P′).

Geometric arguments: If P and P′ are transversal, then Ln(Pδ ∩ P′

δ) ≈ δn−k+1.

If P and P′ are nearly parallel and intersect each other, then Ln(Pδ ∩ P′

δ) ≈ δn−k.

Most of the pairs of δ-tubes intersect in a small enough set. Those pairs which have big intersection are rare enough, namely, for any fixed P′, the amount of affine subspaces P which are nearly parallel to P′ (and intersect it) is small enough.

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 12 / 19

SLIDE 42

L2-estimate and geometric arguments

The Cauchy-Schwarz inequality

E

Ln((P ∩ B)δ)dµ(P) ≤ (Ln(Bδ))1/2 ·

E×E

Ln (Pδ ∩ P′

δ) dµ(P)dµ(P′)

1/2 . We need an upper estimate for

E×E

Ln (Pδ ∩ P′

δ) dµ(P)dµ(P′).

Geometric arguments: If P and P′ are transversal, then Ln(Pδ ∩ P′

δ) ≈ δn−k+1.

If P and P′ are nearly parallel and intersect each other, then Ln(Pδ ∩ P′

δ) ≈ δn−k.

Most of the pairs of δ-tubes intersect in a small enough set. Those pairs which have big intersection are rare enough, namely, for any fixed P′, the amount of affine subspaces P which are nearly parallel to P′ (and intersect it) is small enough. This can be made precise.

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 12 / 19

SLIDE 43

Dimension estimate

It can be proved that for any fixed P′ ∈ E,

E

Ln (Pδ ∩ P′

δ) dµ(P) ≤ δn−k+s log

1 δ

and thus
E×E

Ln (Pδ ∩ P′

δ) dµ(P)dµ(P′) ≤ δn−k+s log

1 δ

,

where s = min(dim E, 1).

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 13 / 19

SLIDE 44

Dimension estimate

It can be proved that for any fixed P′ ∈ E,

E

Ln (Pδ ∩ P′

δ) dµ(P) ≤ δn−k+s log

1 δ

and thus
E×E

Ln (Pδ ∩ P′

δ) dµ(P)dµ(P′) ≤ δn−k+s log

1 δ

,

where s = min(dim E, 1). We get Ln(Bδ) δn−(2α−k+s) log 1

δ

and this immediately implies dim BB ≥ 2α − k + min(dim E, 1).

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 13 / 19

SLIDE 45

Dimension estimate

It can be proved that for any fixed P′ ∈ E,

E

Ln (Pδ ∩ P′

δ) dµ(P) ≤ δn−k+s log

1 δ

and thus
E×E

Ln (Pδ ∩ P′

δ) dµ(P)dµ(P′) ≤ δn−k+s log

1 δ

,

where s = min(dim E, 1). We get Ln(Bδ) δn−(2α−k+s) log 1

δ

and this immediately implies dim BB ≥ 2α − k + min(dim E, 1).

To estimate the Hausdorff dimension of B, we need to find a big enough subset F ⊂ B which is covered by balls of approximately the same radii and estimate Ln(Fδ) from below.

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 13 / 19

SLIDE 46

Future plans

Recall that if F ⊂ R2 is a Furstenberg set, then dim F ≥ 2α and dim F ≥ α + 1

2

by Wolff.

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 14 / 19

SLIDE 47

Future plans

Recall that if F ⊂ R2 is a Furstenberg set, then dim F ≥ 2α and dim F ≥ α + 1

2

by Wolff.

Theorem (H-Keleti-M´ ath´ e)

Let 1 ≤ k < n be integers, and 0 < α ≤ k be any real number. Suppose that B ⊂ Rn, ∅ = E ⊂ A(n, k) and for every k-dimensional affine subspace P ∈ E, dim (P ∩ B) ≥ α. Then dim B ≥ 2α − k + min(dim E, 1). This generalizes the first estimate.

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 14 / 19

SLIDE 48

Future plans

Recall that if F ⊂ R2 is a Furstenberg set, then dim F ≥ 2α and dim F ≥ α + 1

2

by Wolff.

Theorem (H-Keleti-M´ ath´ e)

Let 1 ≤ k < n be integers, and 0 < α ≤ k be any real number. Suppose that B ⊂ Rn, ∅ = E ⊂ A(n, k) and for every k-dimensional affine subspace P ∈ E, dim (P ∩ B) ≥ α. Then dim B ≥ 2α − k + min(dim E, 1). This generalizes the first estimate. Aim: generalize the second estimate.

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 14 / 19

SLIDE 49

Future plans

Recall that if F ⊂ R2 is a Furstenberg set, then dim F ≥ 2α and dim F ≥ α + 1

2

by Wolff.

Theorem (H-Keleti-M´ ath´ e)

Let 1 ≤ k < n be integers, and 0 < α ≤ k be any real number. Suppose that B ⊂ Rn, ∅ = E ⊂ A(n, k) and for every k-dimensional affine subspace P ∈ E, dim (P ∩ B) ≥ α. Then dim B ≥ 2α − k + min(dim E, 1). This generalizes the first estimate. Aim: generalize the second estimate.

Conjecture (H)

Let 1 ≤ k < n be integers, and k − 1 < α ≤ k be any real number. Suppose that B ⊂ Rn, ∅ = E ⊂ A(n, k) and for every k-dimensional affine subspace P ∈ E, dim (P ∩ B) ≥ α. Then dim B ≥ α + dim E

k+1 .

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 14 / 19

SLIDE 50

Future plans

Recall that if F ⊂ R2 is a Furstenberg set, then dim F ≥ 2α and dim F ≥ α + 1

2

by Wolff.

Theorem (H-Keleti-M´ ath´ e)

Let 1 ≤ k < n be integers, and 0 < α ≤ k be any real number. Suppose that B ⊂ Rn, ∅ = E ⊂ A(n, k) and for every k-dimensional affine subspace P ∈ E, dim (P ∩ B) ≥ α. Then dim B ≥ 2α − k + min(dim E, 1). This generalizes the first estimate. Aim: generalize the second estimate.

Conjecture (H)

Let 1 ≤ k < n be integers, and k − 1 < α ≤ k be any real number. Suppose that B ⊂ Rn, ∅ = E ⊂ A(n, k) and for every k-dimensional affine subspace P ∈ E, dim (P ∩ B) ≥ α. Then dim B ≥ α + dim E

k+1 .

If α ≤ k − 1, then this can not hold.

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 14 / 19

SLIDE 51

Future plans

Recall that if F ⊂ R2 is a Furstenberg set, then dim F ≥ 2α and dim F ≥ α + 1

2

by Wolff.

Theorem (H-Keleti-M´ ath´ e)

Let 1 ≤ k < n be integers, and 0 < α ≤ k be any real number. Suppose that B ⊂ Rn, ∅ = E ⊂ A(n, k) and for every k-dimensional affine subspace P ∈ E, dim (P ∩ B) ≥ α. Then dim B ≥ 2α − k + min(dim E, 1). This generalizes the first estimate. Aim: generalize the second estimate.

Conjecture (H)

Let 1 ≤ k < n be integers, and k − 1 < α ≤ k be any real number. Suppose that B ⊂ Rn, ∅ = E ⊂ A(n, k) and for every k-dimensional affine subspace P ∈ E, dim (P ∩ B) ≥ α. Then dim B ≥ α + dim E

k+1 .

If α ≤ k − 1, then this can not hold. Idea: generalize the geometric-combinatorial methods of Wolff.

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 14 / 19

SLIDE 52

Future plans

Recall that if F ⊂ R2 is a Furstenberg set, then dim F ≥ 2α and dim F ≥ α + 1

2

by Wolff.

Theorem (H-Keleti-M´ ath´ e)

Let 1 ≤ k < n be integers, and 0 < α ≤ k be any real number. Suppose that B ⊂ Rn, ∅ = E ⊂ A(n, k) and for every k-dimensional affine subspace P ∈ E, dim (P ∩ B) ≥ α. Then dim B ≥ 2α − k + min(dim E, 1). This generalizes the first estimate. Aim: generalize the second estimate.

Conjecture (H)

Let 1 ≤ k < n be integers, and k − 1 < α ≤ k be any real number. Suppose that B ⊂ Rn, ∅ = E ⊂ A(n, k) and for every k-dimensional affine subspace P ∈ E, dim (P ∩ B) ≥ α. Then dim B ≥ α + dim E

k+1 .

If α ≤ k − 1, then this can not hold. Idea: generalize the geometric-combinatorial methods of Wolff. For k = 1 it is not that hard. For k ≥ 2: research going on.

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 14 / 19

SLIDE 53

Thank you for your attention.

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 15 / 19

SLIDE 54

References

A. Chang, M. Cs¨
rnyei, K. H´

era and T. Keleti, Small unions of affine subspaces and skeletons via Baire category, submitted, arXiv:1701.01405.

K. Falconer and P. Mattila, Strong Marstrand theorems and dimensions of

sets formed by subsets of hyperplanes, J. Fractal Geom. 3 (2016), 319-329.

K. H´

era, T. Keleti , A. M´ ath´ e : Hausdorff dimension of union of affine subspaces, submitted, arXiv:1701.02299.

T. Keleti, Are lines much bigger than line segments?, Proc. Amer. Math.
Soc. 144 (2016), 1535-1541.
T. Keleti, Small union with large set of centers, to appear in Recent

Developments in Fractals and Related Fielsds Conference on Fractals and Related Fields III, ˆ ıle de Porquerolles, France, 2015.

T. Mitsis, On a problem related to sphere and circle packing, J. London
Math. Soc. (2) 60 (1999), 501-516.
T. Wolff, A Kakeya-type problem for circles, Amer. J. Math. 119 (1997),

985-1026.

T. Wolff, Local smoothing type estimates on Lp for large p, Geom. Funct.
Anal. 10 (2000), 1237-1288.
T. Wolff, Lectures on Harmonic Analysis, American Mathematical Society,

University Lecture Series 29 (2003).

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 16 / 19

SLIDE 55

Geometric arguments

Coding of the k-planes

We will use only such affine subspaces that are non-vertical: P = {(t, f (t)) ∈ Rk × Rn−k : t = (t1, . . . , tk) ∈ Rk}, where f (t) = a0 + t1b1 + · · · + tkbk for some a0, b1, . . . , bk ∈ Rn−k. The code of P is defined as (a0, b1, . . . , bk) ∈ R(k+1)(n−k). Put b = max

l=1,...,k

max

j=1,...,n−k |bl j|.

For any small enough δ, if b(P) − b(P′) ≈ mδ for some integer m ∈ {1, . . . , 1

δ},

then Ln(Pδ ∩ P′

δ) δn−k m .

There is a Frostman measure µ on E ⊂ A(n, k) such that µ(B(P, r)) r s for all r > 0 and all P ∈ E, where s = min(dim E, 1). We prove that for any fixed P′ and m ∈ {1, . . . , 1

δ},

µ (P ∈ E : b(P) − b(P′) ≈ mδ, Pδ ∩ P′

δ = ∅) (mδ)s.

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 17 / 19

SLIDE 56

Using an L2-estimate

We need an upper estimate for

E×E

Ln (Pδ ∩ P′

δ) dµ(P)dµ(P′).

It can be proved that for any fixed P′ ∈ E,

E

Ln (Pδ ∩ P′

δ) dµ(P) ≤ log( 1

δ )

j=1
{P∈E:b(P)−b(P′)≈2jδ,Pδ∩P′

δ=∅}

Ln (Pδ ∩ P′

δ) dµ(P)

log( 1

δ )

j=1

δn−k 2j µ

P ∈ E : b(P) − b(P′) ≈ 2jδ, Pδ ∩ P′

δ = ∅

δn−k

log( 1

δ )

j=1

(2jδ)s 2j ≤ δn−k+s log 1 δ

,

using s ≤ 1. Thus Ln(Bδ) δn−(2α−k+s) log 1

δ

and this will imply dim B ≥ 2α − k + min(dim E, 1).

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 18 / 19

SLIDE 57

Minkowski dimension vs Hausdorff dimension

Let B ⊂ ∞

i=1 B(xi, ri) be any countable cover with balls. Our aim is to find a big

enough subset of B which is covered by balls of approximately the same radii and such that many of the affine subspaces of E have big intersection with it. Put Bl = B ∩

{i:ri≈2−l} B(xi, ri).

We have the following by pigeonhole principle: There exists an integer l such that µ

P ∈ E : Hα

∞(P ∩ Bl) ≥ 1

l2

≥ 1

l2 . Putting F =

{P∈E : Hα

∞(P∩Bl)≥ 1 l2 }

(P ∩ Bl) ⊂ B, we can get Ln(Fδ) δn−(2α−k+s) l8 log 1

δ

and this will imply dim B ≥ 2α − k + min(dim E, 1).

Korn´ elia H´ era (E¨

tv¨
s University, Budapest)

Hausdorff dimension of union of affine subspaces Warwick, July 10-14, 2017 19 / 19