The speed of convergence in the renewal theorem

Abstract: In this article we study a diophantine property of probability measures on R. We will always assume that the considered measures have an exponential moment and a drift. We link this property to the points in C close to the imaginary axis where the Fourier-Laplace transform of those measures take the value 1 and finally, we apply this to the study of the speed in Kesten's renewal theorem on R.