zbMATH — the first resource for mathematics

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:
 [1] Krasil’shchikov, V V; Shutov, A V, One-dimensional quasicrystals: approximation by periodic structures and embedding of lattices, 145-154, (2006), Kazan [2] Krasil’shchikov, V V; Shutov, A V, Some questions of the embedding of lattices in one-dimensional quasiperiodic tilings, 84-91, (2007) [3] Krasil’shchikov, V V; Shutov, A V, One-dimensional quasiperiodic tilings admitting progressions enclosure, 3-9, (2009) · Zbl 1195.11084 [4] Krasil’shchikov, V V; Shutov, A V, Distribution of points of one-dimensional quasilattices with respect to a variable module, 17-23, (2012) · Zbl 1347.11017 [5] Krasil’shchikov, V V, The spectrum of one-dimensional quasilattices, Sibirsk. Mat. Zh., 51, 68-73, (2010) · Zbl 1209.11071 [6] Shutov, A V, The arithmetic and geometry of one-dimensional quasilattices, Chebyshevskii Sb., 11, 255-262, (2010) · Zbl 1290.11103 [7] Shutov, A V, Trigonometric sums over one-dimensional quasilattices, Chebyshevskii Sb., 13, 136-148, (2012) · Zbl 1311.11077 [8] I.M. Vinogradov, The Method of Trigonometric Sums in the Theory of Numbers (Nauka, Moscow, 1971; Dover Publications, 2004). · Zbl 0229.10020 [9] Janot, C, quasicrystals: A primer, (1994), Oxford · Zbl 0838.52023 [10] Zhuravlev, V G, Multidimensional Hecke theorem on the distribution of fractional parts, Algebra Anal., 24, 95-130, (2012) [11] Baladi, V; Rockmore, D; Tongring, N; Tresser, C, Renormalization on the $$n$$-dimensional torus, Nonlinearity, 5, 1111-1136, (1992) · Zbl 0761.58008 [12] Rauzy, G, Nombres algébriques et substitutions, Bull. Soc. Math. France, 110, 147-178, (1982) · Zbl 0522.10032 [13] Shutov, A V, The two-dimensional Hecke-Kesten problem, Chebyshevskii Sb., 12, 151-162, (2011) · Zbl 1306.11055 [14] Shutov, A V, On a family of the two-dimensional bounded remainder sets, Chebyshevskii Sb., 12, 264-271, (2011) · Zbl 1302.11048 [15] N. Pytheas Fogg, Substitutions in Dynamics, Arithmetics, and Combinatorics, in Lecture Notes in Math. (Springer-Verlag, Berlin, 2001), Vol. 1794. [16] Zhuravlev, VG, Bounded remainder polyhedra, 82-102, (2012), Moscow [17] Shutov, A V, Multidimensional generalizations of sums of fractional parts and their number-theoretic applications, Chebyshevskii Sb., 14, 104-118, (2013) [18] H. Weyl, “Über die Gibbs’sche Erscheinung und verwandte Konvergenzph änomene,” Palermo Rend. 30, 377-407 (1910). · JFM 41.0528.02 [19] Godrèche, C; Oguey, C, Construction of average lattices for quasiperiodic structures by the section method, J. Phys. France, 51, 21-37, (1990) [20] L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences (Interscience, New York-London-Sydney, 1974; Nauka, Moscow, 1985). · Zbl 0281.10001 [21] Zhuravlev, V G, Fibonacci-even numbers: binary additive problem, distribution over progressions, and spectrum, Algebra Anal., 20, 18-46, (2008) [22] M. Drmota and R. F. Tichy, Sequences, Discrepancies, and Applications, in Lecture Notes in Math. (Springer-Verlag, Berlin, 1997), Vol. 1651. · Zbl 0877.11043 [23] S. A. Stepanov, Arithmetic of Algebraic Curves (Nauka, Moscow, 1991; Springer-Verlag, 1995). · Zbl 0753.14021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.