Notes of Zoltan Balogh’s lecture

1. Projection and slicing in the Heisenberg group

with K. Fässler, P. Mattila, J. Tyson.

We aim at studying complicated objects in metric spaces such as Euclidean space or Heisenberg group. Tomography is a possible approach : cut 2D slices in a 3D object.

1.1. Euclidean projection and slicing

In 1954, Marstrand observed that given a planar set {A} of Hausdorff dimension {t}, {p_\theta (A)} its orthogonal projection to the line {L_\theta} of polar angle {\theta}, {dim(p_\theta(A))} cannot be small for all {\theta}.

Theorem 1 (Marstrand 1954) For almost every {\theta},

\displaystyle  \begin{array}{rcl}  dim(p_\theta(A))= \begin{cases} dim(A) & \text{ if }dim(A) \leq 1, \\ 1 & \text{otherwise}. \end{cases} \end{array}

Furthermore, in the second case, {H^1 (p_\theta(A))>0}.

de Saxcé: there are examples of sets {A} with {dim(A)>1} for which there is a dense {G_{\delta}} of {\theta}‘s such that {dim(p_\theta(A))=0}.

Theorem 2 (Kaufman-Mattila 1975) Let {A} be a subset of {{\mathbb R}^n}. Let {m} be an integer between 1 and {n-1}. For almost every orthogonal projection of rank {m},

\displaystyle  \begin{array}{rcl}  dim(p(A))=\begin{cases} dim(A) & \text{ if }dim(A) \leq m, \\ m & \text{otherwise}. \end{cases} \end{array}

Furthermore, in the second case, {H^m (p(A))>0}.

de Saxcé: This holds for a rather large class of measures on the Grassmannian, satisfying a Hölder type assumption.

A bit more is true: not only does {a+V^\top} intersect {A} for a set of {a}‘s of positive measure in {V}.

Theorem 3 (Mattila’s Slicing Theorem) Let {t=dim(A)>m}. For a.e. {m}-dimensional subspace {V}, for a set of {a}‘s of positive measure in {V}, the intersection of {A} with {a+V^\top} has dimension {\geq t-m}.

1.2. Projection and slicing in the Heisenberg group

The role of {V} will be played by horizontal isotropic subspaces, {W=V^\top} contains the center. These are subgroups, with {W} a normal subgroup. {Heis=WV} is a semi-direct product. The projection {p_V} is a group homomorphism, it is 1-Lipschitz. On the other hand, {p_W} is merely {\frac{1}{2}}-Hölder continuous. The horizontal isotropic Grassmannian is a Riemannian homogeneous space.

Theorem 4 let A be a Borel subset of Heisenberg group. For a.e. {m}-dimensional horizontal isotropic {V}, {dim p_V(A) \leq \min\{m,dim A\}} (easy) but

  1. {dim (p_V(A)) \leq \max\{0,\min\{m,dim(A)-2\}\}}.
  2. Furthermore, if {dim(A)>m+2}, {H^m (p_V(A))>0}.
  3. Let {t=dim(A)>m+2}. Then, for a set of {a}‘s of positive measure in {V}, {dim(Wa \cap A)\geq t-m}.

These bounds are sharp: every pair {(dim A,dim (p_V A))} allowed by these inequalities is achieved.

Remark 1 In particular, in {{\mathbb R}^{2n}}, we have the projection property for a sub-Grassmannian, the isotropic Grassmannian, which is a rather small subspace of the full Grassmannian of {m}-planes in {{\mathbb R}^{2n}}.

de Saxcé: doesn’t it follow from Kaufmann’s estimate on the dimension of the set of bad projections ? Answer: no, Kaufmann’s upper bound is larger than the dimension of the isotropic Grassmannian

Theorem 5 {dim(A) \leq 1 \Rightarrow dim(P_W (A)) \leq 2 dim(A)} (easy) but

For a.e. {m}-dimensional horizontal isotropic {V},

\displaystyle  \begin{array}{rcl}  dim(A) \leq 1 \Rightarrow dim(P_W (A)) \geq dim(A). \end{array}

We do not know what happens when {dim(A)>1}, except for an upper bound in the first Heisenberg group.

Proof relies on the energy characterization of dimension (equality of Hausdorff and capacity dimensions).

About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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