Projections of planar sets in well-separated directions

Abstract

First, let $K\subset B(0,1)\subset {\mathbb{R}}^2$ be a set with ${\mathcal{H}}_{\infty }^1(K)\sim 1$, and write ${\pi }_e(K)$ for the orthogonal projection of $K$ into the line spanned by $e\in S^1$. For $1/2\le s<1$, write $$E_s:=\{e:N({\pi }_e(K),\delta )\le {\delta }^{-s}\},$$ where $N(A,r)$ is the $r$-covering number of the set $A$. It is well-known -- and essentially due to R. Kaufman -- that $N(E_s,\delta )⪅{\delta }^{-s}$. Using the polynomial method, I prove that $$N(E_s,r)⪅\min \left\{{\delta }^{-s}{\left(\frac{\delta }{r}\right)}^{1/2},r^{-1}\right\},\delta \le r\le 1.$$ I construct examples showing that the exponents in the bound are sharp for $\delta \le r\le {\delta }^s$. The second theorem concerns projections of $1$-Ahlfors-David regular sets. Let $A\ge 1$ and $1/2\le s<1$ be given. I prove that, for $p=p(A,s)\in \mathbb{N}$ large enough, the finite set of unit vectors $S_p:=\{e^{2\pi ik/p}:0\le k<p\}$ has the following property. If $K\subset B(0,1)$ is non-empty and $1$-Ahlfors-David regular with regularity constant at most $A$, then $$\frac{1}{p}\sum _{e\in S_p}N({\pi }_e(K),\delta )\ge {\delta }^{-s}$$ for all small enough $\delta >0$. In particular, ${\overline{\dim }}_{\text{B}}{\pi }_e(K)\ge s$ for some $e\in S_p$.

20 pages, 3 figures. v5: Corrected typos and updated references. To appear in Adv. Math

References in corpus (1)

• On the Erdos distinct distance problem in the plane

Cited by in corpus (4)

• On the distance sets of AD-regular sets
• Improving Kaufman's exceptional set estimate for packing dimension
• Planar incidences and geometric inequalities in the Heisenberg group
• Projections in vector spaces over finite fields