| Title:
|
Asymptotics of variance of the lattice point count (English) |
| Author:
|
Janáček, Jiří |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
58 |
| Issue:
|
3 |
| Year:
|
2008 |
| Pages:
|
751-758 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The variance of the number of lattice points inside the dilated bounded set $rD$ with random position in $\Bbb R^d$ has asymptotics $\sim r^{d-1}$ if the rotational average of the squared modulus of the Fourier transform of the set is $O(\rho ^{-d-1})$. The asymptotics follow from Wiener's Tauberian theorem. (English) |
| Keyword:
|
point lattice |
| Keyword:
|
Fourier transform |
| Keyword:
|
volume |
| Keyword:
|
variance |
| MSC:
|
11H06 |
| MSC:
|
62D05 |
| idZBL:
|
Zbl 1174.60002 |
| idMR:
|
MR2455936 |
| . |
| Date available:
|
2010-07-20T14:05:20Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140419 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
[3] Janáček, J.: Variance of periodic measure of bounded set with random position.Comment. Math. Univ. Carolinae 47 (2006), 473-482. MR 2281006 |
| Reference:
|
[4] Kendall, D. G.: On the number of lattice points inside a random oval.Quarterly J. Math. 19 (1948), 1-26. Zbl 0031.11201, MR 0024929, 10.1093/qmath/os-19.1.1 |
| Reference:
|
[5] Kendall, D. G., Rankin, R. A.: On the number of points of a given lattice in a random hypersphere.Quarterly J. Math., 2nd Ser. 4 (1953), 178-189. Zbl 0052.14503, MR 0057484, 10.1093/qmath/4.1.178 |
| Reference:
|
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| Reference:
|
[7] Matheron, G.: Les variables regionalisés et leur estimation.Masson et CIE, Paris (1965). |
| Reference:
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| Reference:
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[9] Rataj, J.: On set covariance and three-point test sets.Czech. Math. J. 54 (2004), 205-214. Zbl 1049.52004, MR 2040232, 10.1023/B:CMAJ.0000027260.34288.7f |
| Reference:
|
[10] Watson, G. N.: A treatise on the theory of Bessel functions.2nd edition, Cambridge University Press (1922). MR 0010746 |
| Reference:
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[11] Rudin, W.: Functional Analysis.McGraw-Hill Book Company (1973). Zbl 0253.46001, MR 0365062 |
| Reference:
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[12] Varchenko, A.: Number of lattice points in families of homothetic domains in $\Bbb R^n$.Func. Anal. Appl. 17 (1983), 79-83. MR 0705041, 10.1007/BF01083133 |
| Reference:
|
[13] Wiener, N.: The Fourier integral and certain of its applications.Dover Publications Inc., New York (1933). Zbl 0006.05401, MR 0100201 |
| . |
