  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.
