Discrete Gaussian measures and new bounds of the smoothing parameter for lattices
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...
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| Published in: | Applicable algebra in engineering, communication and computing Vol. 32; no. 5; pp. 637 - 650 |
|---|---|
| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.11.2021
Springer Nature B.V |
| Subjects: | |
| ISSN: | 0938-1279, 1432-0622 |
| Online Access: | Get full text |
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| Summary: | 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
ε
is suitably small, we obtain two estimations of the smoothing parameter:
η
ε
(
Z
)
≤
ln
(
ε
44
+
2
ε
)
π
.
This is a practically useful case. For this case, our upper bound is very close to the exact value of
η
ε
(
Z
)
in that
ln
(
ε
44
+
2
ε
)
π
-
η
ε
(
Z
)
≤
ε
2
552
.
For a lattice
L
⊂
R
n
of dimension
n
,
η
ε
(
L
)
≤
ln
(
n
-
1
+
2
n
ε
)
π
bl
~
(
L
)
. |
|---|---|
| Bibliography: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0938-1279 1432-0622 |
| DOI: | 10.1007/s00200-020-00417-z |