Lattice point problems and distribution of values of quadratic forms

Bentkus V, Götze F (1999)
ANNALS OF MATHEMATICS 150(3): 977-1027.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Abstract / Bemerkung
For d-dimensional irrational ellipsoids E with d greater than or equal to 9 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)). The estimate refines an earlier authors' bound of order o(r(d-2)) which holds for arbitrary ellipsoids, and is optimal for rational ellipsoids. As an application we prove a conjecture of Davenport and Lewis that the gaps between successive values, say s < n(s), s, n(s) E Q[Zd], of a positive definite irrational quadratic form Q[x], x is an element of R-d, are shrinking, i.e., that n(s)-s -> 0 as s -> infinity, for d greater than or equal to 9. For comparison note that sup,(n(s) - s) < infinity and inf(s)(n(s)-s) > 0, for rational Q[z] and d greater than or equal to 5. As a corollary we derive Oppenheim's conjecture for indefinite irrational quadratic forms, i.e., the set Q[Z(d)] is dense in R, for d greater than or equal to 9, which was proved for d greater than or equal to 3 by G. Margulis [Mar1] in 1986 using other methods. Finally, we provide explicit bounds for errors in terms of certain characteristics of trigonometric sums.
Stichworte
distribution of values of; positive and indefinite quadratic forms; lattice points; quadratic forms; rational and irrational quadratic forms; ellipsoids; Davenport-Lewis conjecture; Oppenheim conjecture
Erscheinungsjahr
1999
Zeitschriftentitel
ANNALS OF MATHEMATICS
Band
150
Ausgabe
3
Seite(n)
977-1027
ISSN
0003-486X
Page URI
https://pub.uni-bielefeld.de/record/1620558

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Bentkus V, Götze F. Lattice point problems and distribution of values of quadratic forms. ANNALS OF MATHEMATICS. 1999;150(3):977-1027.
Bentkus, V., & Götze, F. (1999). Lattice point problems and distribution of values of quadratic forms. ANNALS OF MATHEMATICS, 150(3), 977-1027.
Bentkus, V., and Götze, F. (1999). Lattice point problems and distribution of values of quadratic forms. ANNALS OF MATHEMATICS 150, 977-1027.
Bentkus, V., & Götze, F., 1999. Lattice point problems and distribution of values of quadratic forms. ANNALS OF MATHEMATICS, 150(3), p 977-1027.
V. Bentkus and F. Götze, “Lattice point problems and distribution of values of quadratic forms”, ANNALS OF MATHEMATICS, vol. 150, 1999, pp. 977-1027.
Bentkus, V., Götze, F.: Lattice point problems and distribution of values of quadratic forms. ANNALS OF MATHEMATICS. 150, 977-1027 (1999).
Bentkus, V, and Götze, Friedrich. “Lattice point problems and distribution of values of quadratic forms”. ANNALS OF MATHEMATICS 150.3 (1999): 977-1027.