## 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 $0 \leq r < m_{j_i}, |q m_{j_i}| < C n_{j_i}$

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

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