Quasicircles of dimension 1+k^2 do not exist

Abstract

A well-known theorem of S. Smirnov states that the Hausdorff dimension of a $k$-quasicircle is at most $1+k^2$. Here, we show that the precise upper bound $D(k) = 1+\Sigma^2 k^2 + \mathcal O(k^{8/3-\varepsilon})$ where $\Sigma^2$ is the maximal asymptotic variance of the Beurling transform, taken over the unit ball of $L^\infty$. The quantity $\Sigma^2$ was introduced in a joint work with K. Astala, A. Per\"al\"a and I. Prause where it was proved that $0.879 < \Sigma^2 \le 1$, while recently, H. Hedenmalm discovered that surprisingly $\Sigma^2 <1$. We deduce the asymptotic expansion of $D(k)$ from a more general statement relating the universal bounds for the integral means spectrum and the asymptotic variance of conformal maps. Our proof combines fractal approximation techniques with the classical argument of J. Becker and Ch. Pommerenke for estimating integral means.