We prove a quantitative version of the convergence part of Khintchine’s celebrated theorem in metric Diophantine approximation, but where the approximated points are restricted to lying on the parabola. This talk is based on the preprint https://arxiv.org/pdf/2608.04878 which is joint work with Simon Kristensen.
Unlike other results in literature, we no longer assume the approximation function to be monotonic. This requires us to obtain explicit constants in classical number theoretic results, most notably in Burgess’ bound for short sums of Dirichlet characters for composite moduli.
