Lingmin Liao: Multifractal analysis of the Birkhoff sums of Saint-Petersburg potential

Let $((0,1], T)$ be the doubling map in the unit interval and $\varphi$ be the Saint-Petersburg potential, defined by $\varphi(x)=2^n$ if $x\in (2^{-n-1}, 2^{-n}]$ for all $n\geq 0$.
We consider asymptotic properties of the Birkhoff sum $S_n(x)=\varphi(x)+\cdots+\varphi(T^{n-1}(x))$.
With respect to the Lebesgue measure, the Saint-Petersburg potential is not integrable and it is known that $\frac{1}{n\log n}S_n(x)$ converges to $\frac{1}{\log 2}$ in probability.
We determine the Hausdorff dimension of the level set $\{x: \lim_{n\to\infty}S_n(x)/n=\alpha\} \ (\alpha>0)$, as well as that of the set $\{x: \lim_{n\to\infty}S_n(x)/\Psi(n)=\alpha\} \ (\alpha>0)$, when $\Psi(n)=n\log n$, $n^a $ or $2^{n^\gamma}$ for $a>1$, $\gamma>0$.
This is a joint work with D. H. Kim, M. Rams and B. W. Wang.