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The integer points close to a curve. III. (English) Zbl 0930.11073

Győry, Kálmán (ed.) et al., Number theory in progress. Proceedings of the international conference organized by the Stefan Banach International Mathematical Center in honor of the 60th birthday of Andrzej Schinzel, Zakopane, Poland, June 30–July 9, 1997. Volume 2: Elementary and analytic number theory. Berlin: de Gruyter. 911-940 (1999).
For Part II, cf. Prog. Math. 139, 487-516 (1996; Zbl 0866.11057).
Let \(f(x)\) be a real function on an interval \(I\) of length \(M\geq 1\). Consider the number of lattice points in the strip \(|y-f(x)|\leq\delta\), where \(x\in I\). Estimations of this number are investigated under the assumption that the \(n\)-th derivative \(f^{(n)}(x)\) has a fixed non-zero order of magnitude. The author considers as in Part I [Mathematika 36, 198-215 (1989; Zbl 0659.10032)] the case \(n=2\). There it is assumed that \(\delta\) is very small. In this paper, applying a method of Swinnerton-Dyer, results are obtained which are useful for all \(\delta>0\).
For the entire collection see [Zbl 0911.00018].
Reviewer: E.Krätzel (Wien)

MSC:

11P21 Lattice points in specified regions
11L03 Trigonometric and exponential sums (general theory)
11J25 Diophantine inequalities
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