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Trigonometric sums over one-dimensional quasilattices of arbitrary codimension. (English. Russian original) Zbl 1331.11062
Math. Notes 97, No. 5, 791-802 (2015); translation from Mat. Zametki 97, No. 5, 781-793 (2015).
The author discuss special trigonometric sums of the form \[ f_{n}(\lambda)=\sum_{j=1}^{n}e(x_{j}\lambda). \] Here \(e^{x}\) means \(e^{2\pi i x}\) and \(x_{j}\) is a point of the set of points \(\{x_{n}\}^{\infty}_{-\infty}\) connected with the components \(T_{j}\) of the \(d\)-dimensional torus \(\mathbb{T}^{d}\), \(\mathbb{T}^{d}=T_{0}\sqcup T_{1}\dots \sqcup T_{d}\), discussed by V. G. Zhuravlev [St. Petersbg. Math. J. 24, No. 1, 71–97 (2013; Zbl 1273.11121); translation from Algebra Anal. 24, No. 1, 95–130 (2012)].
The author defines the quasilattice \(Q=Q(\alpha, l_{0}, \dots, l_{d})\), where \(l_{0}, l_{1}, \dots, l_{d}\) are pairwise different positive numbers and considers the same set of points \(\{x_{n}\}^{\infty}_{-\infty}\), given by the conditions \(x_{-1}=0\), \(x_{n+1}=x_{n}+l_{j}\) and \(\alpha = (\alpha_{1},\dots,\alpha_{d})\) is a vector in \(\mathbb{R}^{d}\).
The author proves the following theorem: Let \(h_{Q}=\sum^{d}_{j=0}l_{j}\frac{\mathrm{vol}(T_{j})}{\mathrm{vol}(T)}.\)
1)
If \(h_{Q}\lambda\notin\sum^{d}_{j=1}\alpha_{j}\mathbb{Q}+\mathbb{Q}\), then \(f_{n}(\lambda)=o(n).\)
2)
If \(h_{Q}\lambda=\frac{a_{0}+\sum^{d}_{j=1}a_{j}\alpha_{j}}{b}\), \(a_{0}, a_{1}, \dots,a_{d},b\in\mathbb{Z}\), \((a_{1}, \dots,a_{d},b)=1\), \(|b|>1\), then \(f_{n}(\lambda)=o(n)\).
3)
If \(h_{Q}\alpha =a_{0}+\sum^{d}_{j=1}a_{j}\alpha_{j}\) and \(a_{0}, a_{1}, \dots,a_{d}\in\mathbb{Z}\), then \(f_{n}(\lambda)=c_{Q,\lambda}n+O(\Delta_{2}(\tilde{\alpha},n))\), where \(c_{Q,\lambda}\) is an effectively computable constant and \(\Delta_{2}(\tilde{\alpha},n)\) is explicitly determined in the text.

MSC:
11L03 Trigonometric and exponential sums, general
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