Counting rational points near planar curves
Volume 165 / 2014
Acta Arithmetica 165 (2014), 91-100
MSC: 11J83; 11K60; 11J13.
DOI: 10.4064/aa165-1-5
Abstract
We find an asymptotic formula for the number of rational points near planar curves. More precisely, if $f:\mathbb {R}\rightarrow \mathbb {R}$ 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 $(\xi -\eta )\delta Q^2$.
