On the lattice point problem for ellipsoids

Bentkus, V.; Götze, Friedrich

doi:10.4064/aa-80-2-101-125

Cited by 56 publications

(67 citation statements)

References 12 publications

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“…Finally, for each B ≥ 1 we put N aux f (B) for the number of (d − 1)-tuples of integer n-vectors x (1) , . .…”

Section: ) Misses By a Factor Of O(r)mentioning

confidence: 99%

“…It is in turn founded on work of Bentkus and Götze [1,2] concerning a single quadratic inequality. The method of proof is related to ours, see Sects.…”

Section: Related Workmentioning

confidence: 99%

“…Müller [29] and Bentkus and Götze [1,2] previously used similar ideas to treat quadratic forms with real coefficients. We begin with a technical lemma.…”

Section: Mean Values From Bounds Of the Form (21)mentioning

confidence: 99%

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Quadratic forms and systems of forms in many variables

Myerson

2018

Invent. math.

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Let F 1 , . . . , F R be quadratic forms with integer coefficients in n variables. When n ≥ 9R and the variety V (F 1 , . . . , F R ) is a smooth complete intersection, we prove an asymptotic formula for the number of integer points in an expanding box at which these forms simultaneously vanish, which in particular implies the Hasse principle for V (F 1 , . . . , F R ). Previous work in this direction required n to grow at least quadratically with R. We give a similar result for R forms of degree d, conditional on an upper bound for the number of solutions to an auxiliary inequality. In principle this result may apply as soon as n > d2 d R. In the case that d ≥ 3, several strategies are available to prove the necessary upper bound for the auxiliary inequality. In a forthcoming paper we use these ideas to apply the circle method to nonsingular systems of forms with real coefficients. Mathematics Subject Classification

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“…Finally, for each B ≥ 1 we put N aux f (B) for the number of (d − 1)-tuples of integer n-vectors x (1) , . .…”

Section: ) Misses By a Factor Of O(r)mentioning

confidence: 99%

“…It is in turn founded on work of Bentkus and Götze [1,2] concerning a single quadratic inequality. The method of proof is related to ours, see Sects.…”

Section: Related Workmentioning

confidence: 99%

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Quadratic forms and systems of forms in many variables

Myerson

2018

Invent. math.

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“…Using a linear non-degenerate change of variables, we can transform the operator (1.1) to the form H = HQ + V, Ho = H^ = (DGD) 1 .…”

Section: Definitions and Main Theoremsmentioning

confidence: 99%

“…Also, for d > 5 the estimate (2.11) with k = 0 was shown to hold for rational matrices G (see [14]) and then for arbitrary diagonal matrices G (see [7]). In the recent paper [1] (see also [4]) it was shown that (2.11) and (2.12) (for irrational G^s) hold uniformly in k if d > 9. Thus the conclusions of Proposition 2.4 are valid in dimensions d > 9.…”

Section: Then For Any T > 0 There Exists a (Non-local) Periodic Pertumentioning

confidence: 99%

On the Bethe-Sommerfeld conjecture

Parnovski¹,

Sobolev²

2000

Journées Équations Aux Dérivées Partielles

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Asymptotic Properties of HPD Regions in the Discrete Case

Rousseau

2002

Journal of Multivariate Analysis

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This paper obtains asymptotic expansions of the frequentist distributions of modified likelihood ratio statistics when the observations are discrete. An upper bound of the uncertainty due to the discrete nature of the observations is obtained, which is slightly larger than Yarnold's result (obtained in the case of elliptic confidence regions). Higher order results are also derived from continuity corrections. They are instrumental in the determination of matching priors for HPD regions to the orders o(n −1 ) and O(n −3/2).

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On the lattice point problem for ellipsoids

Cited by 56 publications

References 12 publications

Quadratic forms and systems of forms in many variables

Quadratic forms and systems of forms in many variables

On the Bethe-Sommerfeld conjecture

Asymptotic Properties of HPD Regions in the Discrete Case

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