## 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.

### MSC:

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

### Citations:

Zbl 0691.10022; Zbl 0798.11039
Full Text: