Pointwise equidistribution for almost smooth functions with an error rate and Weighted Lévy-Khintchin theorem

The purpose of this article is twofold: to prove a pointwise equidistribution theorem with an error rate for almost smooth functions, which strengthens the main result of Kleinbock, Shi and Weiss (2017); and to obtain a Lévy-Khintchin theorem for weighted best approximations, which extends the main theorem of Cheung and Chevallier (2019). To do so, we employ techniques from homogeneous dynamics and the methods developed in the work of Cheung-Chevallier (2019) and Shapira-Weiss (2022).