Lattice point problems and values of quadratic forms

Götze F (2004)
INVENTIONES MATHEMATICAE 157(1): 195-226.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Abstract / Bemerkung
For d-dimensional ellipsoids E with dgreater than or equal to5 we show that the number of lattice points in rE is approximated by the volume of rE, as r tends to infinity, up to an error of order O(r(d-2)) for general ellipsoids and up to an error of order o(r(d-2)) for irrational ones. The estimate refines earlier bounds of the same order for dimensions dgreater than or equal to9. As an application a conjecture of Davenport and Lewis about the shrinking of gaps between large consecutive values of Q[m],mis an element ofZ(d) of positive definite irrational quadratic forms Q of dimension dgreater than or equal to5 is proved. Finally, we provide explicit bounds for errors in terms of certain Minkowski minima of convex bodies related to these quadratic forms.
Erscheinungsjahr
2004
Zeitschriftentitel
INVENTIONES MATHEMATICAE
Band
157
Ausgabe
1
Seite(n)
195-226
ISSN
0020-9910
Page URI
https://pub.uni-bielefeld.de/record/1607761

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Götze F. Lattice point problems and values of quadratic forms. INVENTIONES MATHEMATICAE. 2004;157(1):195-226.
Götze, F. (2004). Lattice point problems and values of quadratic forms. INVENTIONES MATHEMATICAE, 157(1), 195-226. doi:10.1007/s00222-004-0366-3
Götze, F. (2004). Lattice point problems and values of quadratic forms. INVENTIONES MATHEMATICAE 157, 195-226.
Götze, F., 2004. Lattice point problems and values of quadratic forms. INVENTIONES MATHEMATICAE, 157(1), p 195-226.
F. Götze, “Lattice point problems and values of quadratic forms”, INVENTIONES MATHEMATICAE, vol. 157, 2004, pp. 195-226.
Götze, F.: Lattice point problems and values of quadratic forms. INVENTIONES MATHEMATICAE. 157, 195-226 (2004).
Götze, Friedrich. “Lattice point problems and values of quadratic forms”. INVENTIONES MATHEMATICAE 157.1 (2004): 195-226.