Geodesic
Estimates of lengths of shortest nonzero vectors in some lattices. I
Diskretnaya Matematika, Tome 33 (2021) no. 1, pp. 31-46.

Voir la notice de l'article provenant de la source Math-Net.Ru

In 1988, Friese et al. put forward lower estimates for the lengths of shortest nonzero vectors for “almost all” lattices of some families in the dimension 3. In 2004, the author of the present paper obtained a similar result for the dimension 4. Here we give a better estimate for the cardinality of the set of exceptional lattices for which the above estimates are not valid. In the case of dimension 4 we improve the upper estimate for an arbitrary chosen parameter that controls the accuracy of these lower estimates and for the number of exceptions. In this (first) part of the paper, we also prove some auxiliary results, which will be used in the second (main) part of the paper, in which an analogue of A. Friese et al. result will be given for dimension 5.
Keywords: lattice, nonzero shortest vectors, Minkowski successive minima. 
@article{DM_2021_33_1_a3,
     author = {A. S. Rybakov},
     title = {Estimates of lengths of shortest nonzero vectors in some lattices. {I}},
     journal = {Diskretnaya Matematika},
     pages = {31--46},
     publisher = {mathdoc},
     volume = {33},
     number = {1},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2021_33_1_a3/}
}
TY  - JOUR
AU  - A. S. Rybakov
TI  - Estimates of lengths of shortest nonzero vectors in some lattices. I
JO  - Diskretnaya Matematika
PY  - 2021
SP  - 31
EP  - 46
VL  - 33
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DM_2021_33_1_a3/
LA  - ru
ID  - DM_2021_33_1_a3
ER  - 
%0 Journal Article
%A A. S. Rybakov
%T Estimates of lengths of shortest nonzero vectors in some lattices. I
%J Diskretnaya Matematika
%D 2021
%P 31-46
%V 33
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DM_2021_33_1_a3/
%G ru
%F DM_2021_33_1_a3
A. S. Rybakov. Estimates of lengths of shortest nonzero vectors in some lattices. I. Diskretnaya Matematika, Tome 33 (2021) no. 1, pp. 31-46. http://geodesic.mathdoc.fr/item/DM_2021_33_1_a3/

[1] Cassels J.W.S., An Introduction to the Geometry of Numbers, Springer-Verlag, 1959 | MR | MR | Zbl

[2] Korn G.A., Korn T.M., Mathematical handbook for scientists and engineers. Definitions, theorems, and formulas for reference and review, McGraw-Hill Book Company, 1968 | MR

[3] Prakhar K., Raspredelenie prostykh chisel, Mir, M., 1967, 512 pp.; Prachar K., Primzahlverteilung, Springer-Verlag, Berlin Göttingen Heidelberg, 1957 | MR | Zbl

[4] Rybakov A.S., “The shortest vectors of lattices connected with a linear congruent generator”, Discrete Math. Appl., 14:5 (2004), 479–500 | DOI | MR | Zbl

[5] Rybakov A.S., “On the number of integer points in a multidimensional domain”, Discrete Math. Appl., 28:6 (2018), 385–395 | DOI | MR | Zbl

[6] Feldman N.I., Sedmaya problema Gilberta, Izd-vo MGU, M., 1982, 312 pp.

[7] Friese A.M., Hastad J., Kannan R., Lagarias J.C., Shamir A., “Reconstructing truncated integer variables satisfying linear congruences”, SIAM J. Comput., 17:2 (1988), 262–280 | DOI | MR