Recurrence is a classical topic in ergodic theory. Poincaré's recurrence theorem (Carathéodory, 1919) says that under some conditions, d(x,Tⁿx) becomes arbitrarily small for almost all points x, that is allmost all points returns arbitrarily close to them-self. Another classical result is by Boshernitzan (1993). It is a more quantitative version of Poincaré's recurrence theorem, and says that if s is the Hausdorff dimension of the space, then, for almost all x, d(x,Tⁿx) is approximately n^-1/s or smaller for infinitely many n.
        In this talk, I will present some new results on recurrence obtain in collaboration with Maxim Kirsebom and Philipp Kunde. For some interval maps and linear maps on tori, we obtained refined scales of recurrence rates compared to Boshernitzan's result. In a very general setting, we have also proved a ''uniform'' version of Boshernitzan's result.