## Posts Tagged ‘Kakeya’

### A few interesting items from the ICM

August 27, 2010

So, as many people know, as part of my India vacation, I went to the ICM in Hyderabad.

On this last day of conference, I decided to write a small note of a few cool items I picked up in some talks: (although there are in fact many, many other cool facts I might write more once I got back :-P)

1. Renormalization (on Artur Avila‘s talk) So we look at one-dimensional systems, one may zoom in at a part of the interval that maps into itself, which yields a similar (or not) system, the idea is called ‘renormalization’. i.e. we have the ‘renormalization operator’ acting on a certain class of systems (‘renormalizable systems’) and this gives a map on the function space. Now we study the dynamics there(!) At the first glance, it doesn’t look like solving a one-dimensional problem in infinite dimensional space would help in any useful way, but it does(!). As an example, we look at space of circle diffeomorphisms, they have a rotation number, it’s not hard to see, in this case, the linear rotations form a circular attractor for the renormalization operator, further more, the dynamics is perfectly understood on the circle (Gauss map), the operator permutes (infinite dimensional) fibers with equal rotational number. It turns out we know enough about the dynamics on the function space to get useful information to the original problem! Super cool~

2. Differentiating Lipschitz functions and decomposing Kakeya sets (on Marianna Csornyei‘s talk, for details please refer to their ICM paper) They had a through study of exactly which sets in $\mathbb{R}^n$ can be contained in the discontinuity set of a Lipschitz function $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$. I found the following unbelievable at the first glance: Given a cone $C$ in $\mathbb{R}^n$ (a set of rays from $\bar{0}$), the $C$-width of set $E \subseteq \mathbb{R}^n$ is, roughly speaking, the $\sup$ of lengths of $E \cap \gamma$ where $\gamma$ is a Lipschitz curve going only in directions in $C$. (A more precise definition requires a generalized notion of ‘tangent’ for Lipschitz curves and can be found in the paper). They proved that:

Theorem: Any Lebesgue $0$ set in $\mathbb{R}^2$ can be decomposed into two sets $A$ and $B$ that $A$ has $C$-width $0$ for $C= [0, \pi/2]$ and $B$ has $C'$-width $0$ for $C' = [\pi/2, \pi]$.

Why does this surprise me? Well, of course the first thing I donsider is: what would happen for the Kakeya set? Our null set contains a line segment in each direction, hence even if we just requiring the decomposed sets $A$ and $B$ to intersect all straight lines in directions of $C, C'$ in length $0$ sets would give pretty much only one possible decomposition: we have to take $A$ to be the union of all segments in direction of $C'$, each missing a linear $0$ set, and same for $B$ and $C$(! not much freedom, right?). It’s already hard to believe such $A$ and $B$ can be made satisfying the property, not to mention that in fact they can be made intersecting all Lipschitz curves in null sets (!) (At first I thought it was an obvious counterexample to the theorem, but after discussing with her after the talk, this is indeed what the theorem does) Amazing…

List to be filled in:

3. Boundary rigidity via filling volume (On Sergei Ivanov‘s talk. For details please refer to his paper on the ArXiv)

4. Constant main curvature surfaces

…to be continued…

### Kaufman’s construction

October 23, 2009

This is a note on R. Kaufman’s paper An exceptional set for Hausdorff dimension

We construct a set $D \subseteq \mathbb{R}^2$ with $\dim(D) = d < 1$ and $E \subseteq [0, \pi)$ with $\dim(E) > 0$ s.t. for all directions $\theta \in E$, $\dim(\pi_\theta(D)) < d-\epsilon$ (the projection of $D$ in direction $\theta$ is less than $d-\epsilon$)

$\forall \alpha >1$, let $(n_j)_{j=1}^\infty$ be an rapidly increasing sequence of integers.

Define $D_j = \{ (a, b)/n_j + \xi \ | \ a, b \in \mathbb{Z}, \ ||(a, b)|| \leq n_j; \ | \xi | \leq n_j^{- \alpha} \}$

i.e. $D_j = \bigcup \{ B((a,b)/n_j, 1/n_j^\alpha) \ | \ (a, b) \in \mathbb{Z}^2 \cap B( \overline{0}, n_j) \}$

Let $D = \bigcap_{j=1}^\infty D_j$

$\because \alpha > 1, \ (n_j)$ rapidly increasing, $\dim(D) = 2 / \alpha$

Let $c \in (0, 1)$ be fixed, define $E' = \{ t \in \mathbb{R} \ | \ \exists$ positive integer sequence $(m_{j_i})_{i=1}^\infty$ s.t. $m_{j_i} < C_1 n_{j_i}^c, \ || m_{j_i} t || < C_2 m_{j_i} / n_{j_i}^\alpha \}$

$\forall t \in E', \ \forall i \in \mathbb{N}, \ \forall p = (a, b)/n_{j_i} + \xi \in D_{j_i}$, we have:

$| \langle p, (1, t) \rangle - a/n_{j_i} - bt/n_{j_i} | \leq (1+|t|)/n_{j_i}^\alpha$

Let $b = q m_{j_i} + r$ where $0 \leq r < m_{j_i}, |q m_{j_i}| < C n_{j_i}$

$\exists z_{j_i} \in \mathbb{Z}, \ | z_{j_i} | < C | n_{j_i} |, \ | \theta |<1$

$bt = qm_{j_i}t +rt = X + rt + q \theta ||m_{j_i}t||$

### A question by Furstenberg

September 25, 2009

Yesterday I was talking about some properties of different dimensions with Furstenburg. Somehow I mentioned Kekaya, and he told me about the following question he has been longing to solve (which is amazingly many similarities to Kekaya):

For set $S \in \mathbb{R}^2$, if $\exists \delta>0$ s.t. for all direction $\theta, \exists$ line $l$ with direction $\theta$ s.t. $\dim (l \cap S) > \delta$. Does it follow that $\dim(A) \geq 1$?

Note that a stronger conjecture would be $\dim(A)$ is at least $1+\delta$ which when taking $\delta = 1$ would give a generalization of the $2$-dimensional Kekaya. (i.e. instead of having to have a line segment, we only require a 1-dimension set in each direction)

Reviewing the proofs of the 2-dimsional Kekaya, I found they all rely on the fact that the line segment is connected…Hence it might be interesting to even find an answer to the following question:

If $A \subseteq \mathbb{R}^2$ contains a measure 1 set in every direction, does it follow that $\dim(A)=2$?