Counting rational points near planar curves

Abstract

We find an asymptotic formula for the number of rational points near planar curves. More precisely, if $f:\mathbb{R}\to \mathbb{R}$ is a sufficiently smooth function defined on the interval $[\eta ,\xi ]$, then the number of rational points with denominator no larger than $Q$ that lie within a $\delta$-neighborhood of the graph of $f$ is shown to be asymptotically equivalent to $(\xi -\eta )\delta Q^2$.

References in corpus (1)

• Diophantine approximation on planar curves: the convergence theory