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Trigonometric integrals over one-dimensional quasilattices of arbitrary codimension. (English. Russian original) Zbl 1359.11071
Math. Notes 99, No. 4, 590-597 (2016); translation from Mat. Zametki 99, No. 4, 603-612 (2016).
The author of this paper studies the one-dimensional quasilattices $$Q=Q(\alpha, l_{1}, l_{2})$$ defined as the set of points $$\{x_{n}\}_{n=-\infty}^{\infty}$$ under the conditions $$x_{-1}=0$$, $$x_{n+1}=\{_{x_{n}+l_{2}, {n\alpha}\geq 1-\alpha.} ^{x_{n}+l_{1}, {n\alpha} <1-\alpha,}$$, where $$l_{1}, l_{2}$$ are arbitrary distinct positive numbers, $$\alpha$$ is irrational number. This paper is a continuation and generalization of the previous paper [the author, Math. Notes 97, No. 5, 791–802 (2015; Zbl 1331.11062); translation from Mat. Zametki 97, No. 5, 781–793 (2015)]. The author studies trigonometric integrals of the form $$I_{n,d}=\int_{\mathbb{T}^{d}}\int_{0}^{1}|f_{n}(\alpha, \lambda)|d\lambda d\alpha$$, where $$\mathbb{T}^{d}$$ is, as in the previous paper, exchanged tiling of the $$d$$-dimensional torus $$\mathbb{T}^{d}=T_{0}\bigsqcup T_{1}\bigsqcup \cdots \bigsqcup T_{d}$$. The main result is the following theorem: Let $$\varphi$$ be a monotone increasing function such that $$\varphi>0$$ for all positive integer $$n$$ and the series $$\Sigma_{n=1}^{\infty}1/\varphi(n)$$ converges. Then the following estimate holds: $$I_{n,d}=O(\ln^{d+1}n \varphi(\ln \ln n))$$. The paper ends with a posed problem about an inequality in the theory of Diophantine approximations.
##### MSC:
 11L03 Trigonometric and exponential sums, general 11H31 Lattice packing and covering (number-theoretic aspects)
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##### References:
 [1] C. Janot, Quasicrystals: A Primer (Clarendon Press, Oxford, 1994). · Zbl 0838.52023 [2] Krasil’shchikov, V. V.; Shutov, A. V., One-dimensional quasicrystals: approximation by periodic structures and enclosure of lattices, 145-154, (2006) [3] Krasil’shchikov, V. V.; Shutov, A. V., Several problems of enclosure of lattices in one-dimensional quasiperiodic tilings, Vestnik Samara Gos. Univ., Estestvennonauch. Ser., 7, 84-91, (2007) [4] Krasil’shchikov, V. V.; Shutov, A. V., One-dimensional quasiperiodic tilings admitting progressions enclosure, Izv. Vyssh. Uchebn. Zaved. Mat., 7, 3-9, (2009) · Zbl 1195.11084 [5] Krasil’shchikov, V. V.; Shutov, A. V., Distribution of points of one-dimensional quasilattices with respect to a variable module, Izv. Vyssh. Uchebn. Zaved. Mat., 3, 17-23, (2012) · Zbl 1347.11017 [6] Krasil’shchikov, V. V., The spectrum of one-dimensional quasilattices, Sibirsk. Mat. Zh., 51, 68-73, (2010) · Zbl 1209.11071 [7] Shutov, A. V., The arithmetic and geometry of one-dimensional quasilattices, Chebyshevskii Sb., 11, 255-262, (2010) · Zbl 1290.11103 [8] Shutov, A. V., Trigonometric sums over one-dimensional quasilattices, Chebyshevskii Sb., 13, 136-148, (2012) · Zbl 1311.11077 [9] Shutov, A. V., Trigonometric sums over one-dimensional quasilattices of arbitrary codimension, Mat. Zametki, 97, 781-793, (2015) · Zbl 1331.11062 [10] I. M. Vinogradov, The Method of Trigonometric Sums in the Theory of Numbers (Nauka, Moscow, 1971; Dover Publ., 2004). [11] Zhuravlev, V. G., A multidimensional Hecke theorem on the distribution of fractional parts, Algebra Anal., 24, 95-130, (2012) [12] Baladi, V.; Rockmore, D.; Tongring, N.; Tresser, C., Renormalization on the n-dimensional torus, Nonlinearity, 5, 1111-1136, (1992) · Zbl 0761.58008 [13] Rauzy, G., Nombres algébriques et substitutions, Bull. Soc. Math. France, 110, 147-178, (1982) · Zbl 0522.10032 [14] Shutov, A. V., The two-dimensional Hecke-Kesten problem, Chebyshevskii Sb., 12, 151-162, (2011) · Zbl 1306.11055 [15] Shutov, A. V., On a family of two-dimensional bounded remainder sets, Chebyshevskii Sb., 12, 264-271, (2011) · Zbl 1302.11048 [16] Abrosimova, A. A., Bounded remainder sets on a two-dimensional torus, Chebyshevskii Sb., 12, 15-23, (2011) · Zbl 1306.11054 [17] N. Pytheas Fogg, Substitutions in Dynamics, Arithmetics and Combinatorics (Springer-Verlag, Berlin, 2002). · Zbl 1014.11015 [18] Zhuravlev, V. G., Bounded remainder polyhedra, 82-102, (2012) [19] Abrosimova, A. A., BR-sets, Chebyshevskii Sb., 16, 8-22, (2015) [20] Shutov, A. V., Multidimensional generalizations of sums of fractional parts and their number-theoretic applications, Chebyshevskii Sb., 14, 104-118, (2013) [21] Weyl, H., Über die gibbs’sche erscheinung und verwandte konvergenzphänomene, Rend. Circ. Math. Palermo, 30, 377-407, (1910) · JFM 41.0528.02 [22] L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences (Interscience, New York-London-Sydney, Mir, Moscow, 1985). · Zbl 0568.10001 [23] Beck, J., Probabilistic Diophantine approximation. I. Kronecker sequences, Ann. Math., 140, 451-502, (1994) · Zbl 0820.11045 [24] Drmota, M.; Tichy, R. F., Sequences, discrepancies and applications, (1997) · Zbl 0877.11043
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