Trigonometric polynomials and lattice points. (English) Zbl 0777.11035

The authors study the distribution of lattice points on comparatively small portions of large circles (*) \(x^ 2+y^ 2=R^ 2\), motivated by an observation attributed to A. Schinzel that Heron’s theorem from elementary geometry implies that any arc of length \(\leq 2R^{1/3}\) of the circle (*) contains at most two lattice points.
By an ingenious new approach, the authors are able to establish the following general result which says that lattice points cannot lie “too dense” on such a circle:
Theorem 1. For an integer \(m\geq 2\), let \(\alpha(m)={1\over 2} - {1 \over {4[{m\over 2}]+2}}\). Then any arc of the circle (*) whose length does not exceed \(\sqrt{2} R^{\alpha(m)}\) contains at most \(m\) lattice points.
By means of the analysis carried out, the authors obtain a precise \(L^ 4\)-asymptotic for (“short”) incomplete Gaussian sums:
Theorem 2. For large \(N\), fixed \(\alpha\) with \({\textstyle {1\over 2}}<\alpha<1\), and every \(\varepsilon>0\), \[ \int^ 1_ 0 \bigl| \sum_{N\leq k\leq N+N^ \alpha} e^{2\pi ik^ 2 x} \bigr|^ 4 dx =2N^{2\alpha}+O(N^{3\alpha-1+\varepsilon}). \] According to the reviewer’s opinion, this paper is an ingenious and remarkable contribution to recent lattice point theory.
Reviewer: W.G.Nowak (Wien)


11P21 Lattice points in specified regions
11L05 Gauss and Kloosterman sums; generalizations
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