Archive for October, 2009

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||