Multifractal analysis of the Birkhoff sums of Saint-Petersburg potential

Let $((0,1],T)$ be the doubling map in the unit interval and $\phi$ be the Saint-Petersburg potential, defined by $\phi (x)=2^n$ if $x\in (2^{-n-1}, 2^{-n}]$forall$n\geq 0$. We consider the asymptotic properties of the Birkhoff sum $S\mathrm{\_}n(x)=\phi (x)+\cdots +\phi (T^{n-1}(x))$. With respect to the Lebesgue measure, the Saint-Petersburg potential is not integrable and it is known that $\frac{1}{n\log n}S\mathrm{\_}n(x)$ converges to $\frac{1}{\log 2}$ in probability. We determine the Hausdorff dimension of the level set ${x: \lim_{n\to\infty}S_n(x)/n=\alpha} \ (\alpha>0)$,aswellasthatoftheset${x: \lim_{n\to\infty}S_n(x)/\Psi(n)=\alpha} \ (\alpha>0)$,when$\Psi(n)=n\log n, n^a $or$2^{n^\gamma}$for$a>1$,$\gamma>0$. The fast increasing Birkhoff sum of the potential function $x\mapsto 1\mathrm{/}x$ is also studied.