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Erschienen in: Applicable Algebra in Engineering, Communication and Computing 5/2021

07.02.2020 | Original Paper

Discrete Gaussian measures and new bounds of the smoothing parameter for lattices

verfasst von: Zhongxiang Zheng, Chunhuan Zhao, Guangwu Xu

Erschienen in: Applicable Algebra in Engineering, Communication and Computing | Ausgabe 5/2021

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Abstract

In this paper, we start with a discussion of discrete Gaussian measures on lattices. Several results of Banaszczyk are analyzed, a simple form of uncertainty principle for discrete Gaussian measure is formulated. In the second part of the paper we prove two new bounds for the smoothing parameter of lattices. Under the natural assumption that \(\varepsilon\) is suitably small, we obtain two estimations of the smoothing parameter:
1.
$$\begin{aligned} \displaystyle \eta _{\varepsilon }({{\mathbb {Z}}}) \le \sqrt{\frac{\ln \big (\frac{\varepsilon }{44}+\frac{2}{\varepsilon }\big )}{\pi }}. \end{aligned}$$
This is a practically useful case. For this case, our upper bound is very close to the exact value of \(\eta _{\varepsilon }({{\mathbb {Z}}})\) in that \(\sqrt{\frac{\ln \big (\frac{\varepsilon }{44}+\frac{2}{\varepsilon }\big )}{\pi }}-\eta _{\varepsilon }({{\mathbb {Z}}})\le \frac{\varepsilon ^2}{552}\).
 
2.
For a lattice \({{{\mathcal {L}}}}\subset {{\mathbb {R}}}^n\) of dimension n,
$$\begin{aligned} \displaystyle \eta _{\varepsilon }({{{\mathcal {L}}}}) \le \sqrt{\frac{\ln \big (n-1+\frac{2n}{\varepsilon }\big )}{\pi }}\tilde{bl}({{{\mathcal {L}}}}). \end{aligned}$$
 

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Fußnoten
1
This means that f and all its (partial) derivatives \(D^{\beta }f\) are rapidly decreasing in the sense that \(\sup _{{\mathbf {x}}\in {{\mathbb {R}}}^n} |{\mathbf {x}}^{\alpha }D^{\beta }f({\mathbf {x}})|<\infty\) for every \(\alpha , \beta \in {{\mathbb {N}}}^n\). Such a function is said to be in the Schwartz space.
 
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Metadaten
Titel
Discrete Gaussian measures and new bounds of the smoothing parameter for lattices
verfasst von
Zhongxiang Zheng
Chunhuan Zhao
Guangwu Xu
Publikationsdatum
07.02.2020
Verlag
Springer Berlin Heidelberg
Erschienen in
Applicable Algebra in Engineering, Communication and Computing / Ausgabe 5/2021
Print ISSN: 0938-1279
Elektronische ISSN: 1432-0622
DOI
https://doi.org/10.1007/s00200-020-00417-z

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