Systems of quadratic diophantine inequalities

Wolfgang Müller

doi:10.5802/jtnb.488

Wolfgang Müller ¹

¹ Institut für Statistik Technische Universität Graz 8010 Graz, Austria

Journal de théorie des nombres de Bordeaux, Volume 17 (2005) no. 1, pp. 217-236.

Abstract
Résumé

Let $Q_{1}, \dots, Q_{r}$ be quadratic forms with real coefficients. We prove that for any $ϵ > 0$ the system of inequalities $| Q_{1} (x) | < ϵ, \dots, | Q_{r} (x) | < ϵ$ has a nonzero integer solution, provided that the system $Q_{1} (x) = 0, \dots, Q_{r} (x) = 0$ has a nonsingular real solution and all forms in the real pencil generated by $Q_{1}, \dots, Q_{r}$ are irrational and have rank $> 8 r$ .

Soient $Q_{1}, ..., Q_{r}$ des formes quadratiques avec des coefficients réels. Nous prouvons que pour chaque $ε > 0$ le système $| Q_{1} (x) | < ε, ..., | Q_{r} (x) | < ε$ des inégalités a une solution entière non-triviale si le système $Q_{1} (x) = 0, ..., Q_{r} (x) = 0$ a une solution réelle non-singulière et toutes les formes $\sum_{i = 1}^{r} α_{i} Q_{i}$ , $α = (α_{1}, ..., α_{r}) \in ℝ^{s}, α \neq 0$ sont irrationnelles avec rang $> 8 r$ .

MR Zbl

DOI: 10.5802/jtnb.488

Author's affiliations:

Wolfgang Müller ¹

¹ Institut für Statistik Technische Universität Graz 8010 Graz, Austria

@article{JTNB_2005__17_1_217_0,
     author = {Wolfgang M\"uller},
     title = {Systems of quadratic diophantine inequalities},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {217--236},
     publisher = {Universit\'e Bordeaux 1},
     volume = {17},
     number = {1},
     year = {2005},
     doi = {10.5802/jtnb.488},
     mrnumber = {2152222},
     zbl = {1082.11020},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.488/}
}

TY  - JOUR
AU  - Wolfgang Müller
TI  - Systems of quadratic diophantine inequalities
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2005
SP  - 217
EP  - 236
VL  - 17
IS  - 1
PB  - Université Bordeaux 1
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.488/
DO  - 10.5802/jtnb.488
LA  - en
ID  - JTNB_2005__17_1_217_0
ER  -

%0 Journal Article
%A Wolfgang Müller
%T Systems of quadratic diophantine inequalities
%J Journal de théorie des nombres de Bordeaux
%D 2005
%P 217-236
%V 17
%N 1
%I Université Bordeaux 1
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.488/
%R 10.5802/jtnb.488
%G en
%F JTNB_2005__17_1_217_0

Wolfgang Müller. Systems of quadratic diophantine inequalities. Journal de théorie des nombres de Bordeaux, Volume 17 (2005) no. 1, pp. 217-236. doi : 10.5802/jtnb.488. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.488/

References
Cited by

[1] V. Bentkus, F. Götze, On the lattice point problem for ellipsoids. Acta Arith. 80 (1997), 101–125. | MR | Zbl

[2] V. Bentkus, F. Götze, Lattice point problems and distribution of values of quadratic forms. Annales of Mathematics 150 (1999), 977–1027. | MR | Zbl

[3] E. Bombieri, H. Iwaniec, On the order of $ζ (1 / 2 + i t)$ . Ann. Scuola Norm Sup. Pisa (4) 13 (1996), 449–472. | Numdam | Zbl

[4] J. Brüdern, R.J. Cook, On simultaneous diagonal equations and inequalities. Acta Arith. 62 (1992), 125–149. | MR | Zbl

[5] H. Davenport, Indefinite quadratic forms in many variables (II). Proc. London Math. Soc. 8 (1958), 109–126. | MR | Zbl

[6] R. J. Cook, Simultaneous quadratic equations. Journal London Math. Soc. (2) 4 (1971), 319–326. | MR | Zbl

[7] R. Dietmann, Systems of rational quadratic forms. Arch. Math. (Basel) 82 (2004), no. 6, 507–516. | MR | Zbl

[8] E. D. Freeman, Quadratic Diophantine Inequalities. Journal of Number Theory 89 (2001), 269–307. | MR | Zbl

[9] G. A. Margulis, Discrete subgroups and ergodic theory. In ”Number Theory, Trace Fromulas and Discrete Groups (Oslo, 1987)”, 377–398, Academic Press, Boston, 1989. | MR | Zbl

[10] W. M. Schmidt, Simultaneous rational zeros of quadratic forms. Séminaire Delange-Pisot-Poitou (Théorie des Nombres), Paris 1980-1981, Progr. Math. 22 (1982), 281–307. | MR | Zbl

[11] H. P. F. Swinnerton-Dyer, Rational zeros of two quadratic forms. Acta Arith. 9 (1964), 261–270. | MR | Zbl

Cited by Sources: