Vitushkin's conjecture

For an open set $U\subseteq\mathbf{C}$, let $\mathcal{H}^{\infty}(U)$ denote the set of bounded analytic functions $f\colon U\to\mathbf{C}$, and for $f$ analytic near $\infty$ put

$$f(\infty):=\lim_{z\to\infty}f(z),\qquad f'(\infty):=\lim_{z\to\infty}z\bigl(f(z)-f(\infty)\bigr).$$

For a compact set $K\subset\mathbf{C}$, the analytic capacity of $K$ is

$$\gamma(K)=\sup\{|f'(\infty)|\;:\;f\in\mathcal{H}^{\infty}(\mathbf{C}\setminus K),\ \|f\|_{\infty}\leq 1,\ f(\infty)=0\}.$$

Let $\mathcal{H}^{1}$ denote $1$-dimensional Hausdorff measure, and for $\theta\in[0,\pi)$ let

$$\operatorname{proj}_{\theta}(x,y)=x\cos\theta+y\sin\theta$$

be the orthogonal projection of $\mathbf{C}\cong\mathbf{R}^{2}$ onto the line through the origin in direction $\theta$.

For every compact set $K\subset\mathbf{C}$,

$$\gamma(K)=0\quad\Longleftrightarrow\quad\int_{0}^{\pi}\mathcal{H}^{1}\bigl(\operatorname{proj}_{\theta}(K)\bigr)\,d\theta=0 .$$