# zbMATH — the first resource for mathematics

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)
Full Text:
##### References:
  C. Janot, Quasicrystals: A Primer (Clarendon Press, Oxford, 1994). · Zbl 0838.52023  Krasil’shchikov, V. V.; Shutov, A. V., One-dimensional quasicrystals: approximation by periodic structures and enclosure of lattices, 145-154, (2006)  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)  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  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  Krasil’shchikov, V. V., The spectrum of one-dimensional quasilattices, Sibirsk. Mat. Zh., 51, 68-73, (2010) · Zbl 1209.11071  Shutov, A. V., The arithmetic and geometry of one-dimensional quasilattices, Chebyshevskii Sb., 11, 255-262, (2010) · Zbl 1290.11103  Shutov, A. V., Trigonometric sums over one-dimensional quasilattices, Chebyshevskii Sb., 13, 136-148, (2012) · Zbl 1311.11077  Shutov, A. V., Trigonometric sums over one-dimensional quasilattices of arbitrary codimension, Mat. Zametki, 97, 781-793, (2015) · Zbl 1331.11062  I. M. Vinogradov, The Method of Trigonometric Sums in the Theory of Numbers (Nauka, Moscow, 1971; Dover Publ., 2004).  Zhuravlev, V. G., A multidimensional Hecke theorem on the distribution of fractional parts, Algebra Anal., 24, 95-130, (2012)  Baladi, V.; Rockmore, D.; Tongring, N.; Tresser, C., Renormalization on the n-dimensional torus, Nonlinearity, 5, 1111-1136, (1992) · Zbl 0761.58008  Rauzy, G., Nombres algébriques et substitutions, Bull. Soc. Math. France, 110, 147-178, (1982) · Zbl 0522.10032  Shutov, A. V., The two-dimensional Hecke-Kesten problem, Chebyshevskii Sb., 12, 151-162, (2011) · Zbl 1306.11055  Shutov, A. V., On a family of two-dimensional bounded remainder sets, Chebyshevskii Sb., 12, 264-271, (2011) · Zbl 1302.11048  Abrosimova, A. A., Bounded remainder sets on a two-dimensional torus, Chebyshevskii Sb., 12, 15-23, (2011) · Zbl 1306.11054  N. Pytheas Fogg, Substitutions in Dynamics, Arithmetics and Combinatorics (Springer-Verlag, Berlin, 2002). · Zbl 1014.11015  Zhuravlev, V. G., Bounded remainder polyhedra, 82-102, (2012)  Abrosimova, A. A., BR-sets, Chebyshevskii Sb., 16, 8-22, (2015)  Shutov, A. V., Multidimensional generalizations of sums of fractional parts and their number-theoretic applications, Chebyshevskii Sb., 14, 104-118, (2013)  Weyl, H., Über die gibbs’sche erscheinung und verwandte konvergenzphänomene, Rend. Circ. Math. Palermo, 30, 377-407, (1910) · JFM 41.0528.02  L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences (Interscience, New York-London-Sydney, Mir, Moscow, 1985). · Zbl 0568.10001  Beck, J., Probabilistic Diophantine approximation. I. Kronecker sequences, Ann. Math., 140, 451-502, (1994) · Zbl 0820.11045  Drmota, M.; Tichy, R. F., Sequences, discrepancies and applications, (1997) · Zbl 0877.11043
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.