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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
Full Text:
##### References:
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