Multifractal Analysis of the Birkhoff Sums of Saint-Petersburg Potential

Let ((0, 1],T) be the doubling map in the unit interval and φ be the Saint-Petersburg potential, defined by φ(x) = 2n if x ∈ (2-n-1, 2-n] for all n ≥ 0. We consider asymptotic properties of the Birkhoff sum Sn(x) = φ(x) + ⋯ + φ(Tn-1(x)). With respect to the Lebesgue measure, the Saint-Petersburg potential is not integrable and it is known that 1 nlog nSn(x) converges to 1 log 2 in probability. We determine the Hausdorff dimension of the level set {x:limn→∞Sn(x)/n = α} (α > 0), as well as that of the set {x:limn→∞Sn(x)/Ψ(n) = α} (α > 0), when Ψ(n) = nlog n, na or 2nγ for a > 1,γ > 0. The fast increasing Birkhoff sum of the potential function x↦1/x is also studied.