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Bombieri and Pila had a well known bound for the count of lattice points on an algebraic curve in the plane. Does it generalize to a bound for the count of lattice points near (say within a distance of $\delta=o(1)$) an algebraic curve? Is it obvious that we only need to add $O(\delta L)$ to the Bombieri-Pila count, where $L$ is the length of the curve? Or do we have to use the weaker bound of Swinnerton-Dyer?

Edit: To put things in more context, $O(\delta L)$ is the "expected" number of lattice points near the curve if the curve is situated randomly in the plane. The Bombieri-Pila bound can be thought of as arising from the arithmetic structure of the algebraic curve. My question then amounts to: do the expected count and the arithmetic count together fully account for the lattice points near an algebraic curve?

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  • $\begingroup$ what is relation between $\delta$ and $L$? $\endgroup$ – Fedor Petrov Jan 26 '16 at 0:48
  • $\begingroup$ @FedorPetrov Assume $\delta$ is very small. $\endgroup$ – Fan Zheng Jan 26 '16 at 1:54
  • $\begingroup$ See Huxley's paper: Huxley, M. N. The integer points close to a curve. Mathematika 36 (1989), no. 2, 198–215 (1990). (Reviewer: S. W. Graham) 11J54 (11J71) This type of problem can also be approached by an extension of the Bombieri-Pila determinant method, due to Heath-Brown. This version is called the "approximate" determinant method. For example, see: blms.oxfordjournals.org/content/47/2/270 and Heath-Brown, D. R. Sums and differences of three kth powers. J. Number Theory 129 (2009), no. 6, 1579–1594 $\endgroup$ – Stanley Yao Xiao Jan 29 '16 at 22:25
  • $\begingroup$ Also, Jing Jing Huang does a lot of work on this topic; searching his name on MathSciNet will likely be fruitful. $\endgroup$ – Stanley Yao Xiao Jan 29 '16 at 22:27
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See: MR2271609 (2007m:11092) Reviewed Huxley, Martin N.(4-CARD-SM); Sargos, Patrick(F-NANC-IE) Points entiers au voisinage d'une courbe plane de classe Cn. II. (French. English, French summary) [Integral points near a plane curve of class Cn. II] Funct. Approx. Comment. Math. 35 (2006), 91–115. 11J25 (11P21)

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  • $\begingroup$ Is it about algebraic or smooth curves? $\endgroup$ – Fedor Petrov Jan 29 '16 at 23:38
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    $\begingroup$ @FedorPetrov Actually both are helpful to me, but what I need most now is a French dictionary :). $\endgroup$ – Fan Zheng Jan 29 '16 at 23:39
  • $\begingroup$ @FanZheng There are more symbols than words there :) And yes, the paper is about smooth curves. $\endgroup$ – Igor Rivin Jan 30 '16 at 8:50

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