A near-optimal solution to the Gauss-Kuzmin-Lévy problem for θ-expansions

Chakraborty and Rao [4] considered the θ-expansions of numbers in [0,θ), where 0<θ<1. A Wirsing-type approach to the Perron-Frobenius operator of the generalized Gauss map under its invariant measure allows us to study the optimality of the convergence rate. Actually, we obtain upper and lower bounds of the convergence rate which provide a near-optimal solution to the Gauss-Kuzmin-Lévy problem.