Diophantine approximation on the parabola with non-monotonic approximation functions

TL;DR

This paper proves that the parabola exhibits strong Khintchine type convergence properties and establishes Jarnik type theorems in both settings without requiring monotonicity of approximation functions, using advanced counting techniques.

Contribution

It is the first to demonstrate strong Khintchine type convergence for the parabola and extends Jarnik theorems without monotonicity assumptions, employing new counting methods.

Findings

Parabola is of strong Khintchine type for convergence

Jarnik type theorems are established without monotonicity

A new counting result for rational points near the parabola

Abstract

We show that the parabola is of strong Khintchine type for convergence, which is the first result of its kind for curves. Moreover, Jarnik type theorems are established in both the simultaneous and the dual settings, without monotonicity on the approximation function. To achieve the above, we prove a new counting result for the number of rational points with fixed denominators lying close to the parabola, which uses Burgess's bound on short character sums.