LAUSR

Let [Formula: see text] be the doubling map in the unit interval and [Formula: see text] be the Saint-Petersburg potential, defined by [Formula: see text] if [Formula: see text] for all [Formula: see text]. We consider asymptotic properties of the Birkhoff sum [Formula: see text]. With respect to the Lebesgue measure, the Saint-Petersburg potential is not integrable and it is known that [Formula: see text] converges to [Formula: see text] in probability. We determine the Hausdorff dimension of the level set [Formula: see text], as well as that of the set [Formula: see text], when [Formula: see text], [Formula: see text] or [Formula: see text] for [Formula: see text]. The fast increasing Birkhoff sum of the potential function [Formula: see text] is also studied.