Sinai initiated the study of random walks with persistently positive area processes. We find the precise asymptotic probability that the area process of a random walk bridge is an excursion. The asymptotics are related to subset counting formulas from additive number theory. Our results respond to a question of Caravenna and Deuschel, which arose in the context of the wetting model from statistical physics. Joint works with Michal Bassan and Serte Donderwinkel will be discussed.