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Counting rational points near planar curves

Volume 165 / 2014

Ayla Gafni Acta Arithmetica 165 (2014), 91-100 MSC: 11J83; 11K60; 11J13. DOI: 10.4064/aa165-1-5


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 $(\xi -\eta )\delta Q^2$.


  • Ayla GafniDepartment of Mathematics
    Pennsylvania State University
    109 McAllister Bldg
    University Park, PA 16802, U.S.A.

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