an:06640396
Zbl 1359.11071
Shutov, A. V.
Trigonometric integrals over one-dimensional quasilattices of arbitrary codimension
EN
Math. Notes 99, No. 4, 590-597 (2016); translation from Mat. Zametki 99, No. 4, 603-612 (2016).
00356828
2016
j
11L03 11H31
quasilattice; trigonometric integrals; trigonometric sums; Diophantine approximation; exchanged tiling of \(d\)-dimensional torus
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.
Tonko Tonkov (Sofia)
Zbl 1331.11062