Integer points close to a plane curve of class \(C^ n\). (Points entiers au voisinage d’une courbe plane de classe \(C^ n\).) (French) Zbl 0822.11070

Let \(\Gamma\) be a \(C^ n\) curve given by \(y= f(x)\) for \(a\leq x\leq a+N\), and let \(\Gamma_ \delta\) be the strip of points \((x, y+t)\) for \((x,y)\) on \(\Gamma\), \(| t|\leq \delta\). Here \(N\geq 2\) is an integer, and \(\delta< 1/2\) is non-negative. Exponential sum methods express \(\nu\), the number of integer points in \(\Gamma_ \delta\), as \(2\delta N\) plus an error term involving estimates for the derivatives; when \(\delta\) is small, the result becomes an upper bound only. The first author [Mathematika 36, 198-215 (1989; Zbl 0691.10022)] obtained an upper bound for \(\nu\) on the sole assumption that \(| f^{(n)} (x)| \asymp \lambda\), with \(0< \lambda<1\). M. Branton and the second author [Bull. Sci. Math., II. Ser. 118, 15-28 (1994; Zbl 0798.11039)] considered the case \(n=2\) in detail.
The earlier results are improved to \[ \nu\ll N\lambda^{2/ n(n+1)}+ N\delta^{2/ n(n-1)}+ (\delta\lambda^{-1})^{1/n}+ 1, \] by considering polynomial spline approximations to \(f(x)\) with rational coefficients. The “major arcs”, good approximations with small denominator, require the most work.


11P21 Lattice points in specified regions
11K99 Probabilistic theory: distribution modulo \(1\); metric theory of algorithms
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