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Lattice points. (English) Zbl 0910.11042

The author states a theorem due to J. Cilleruelo and A. Córdoba [Duke Math. J. 76, 741-750 (1994; Zbl 0822.11069)]. Consider a circle of radius \(R\) centered at the origin. Then an arc of length \(R^\alpha\) contains, at most, \(c_\alpha\) lattice points, where \(1/3\leq \alpha\leq 1/2\) and \(c_\alpha\) is a finite constant.
Then some asymptotic results on Gauss sums are given and the differentiability of the trigonometric series \[ S_{\alpha,k} (x)= \sum^\infty_{n=1}{1\over n^\alpha} e^{2 \pi in^kx} \] is discussed. Finally, examples for sums of type \[ S(N)= \sum^N_{k=1} f\left( {k\over N} \right) \mu \left(N\varphi \left( {k\over N} \right) \right) \] are given, where \(\mu\) is a periodic function of average 0 and \(| \varphi''(x) |\geq c>0\).
Reviewer: E.Krätzel (Wien)

MSC:

11P21 Lattice points in specified regions
11H06 Lattices and convex bodies (number-theoretic aspects)
11L05 Gauss and Kloosterman sums; generalizations

Citations:

Zbl 0822.11069
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References:

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