A generic effective Oppenheim theorem for systems of forms

Bandi, Prasuna; Ghosh, Anish; Han, Jiyoung

doi:10.1016/j.jnt.2020.07.002

Cited by 11 publications

(7 citation statements)

References 16 publications

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“…We also mention the work of Bourgain [7] on certain "uniform" versions of Oppenheim's conjecture for diagonal forms. See also [13] for an analogue for ternary forms and [5,20].…”

Section: Introductionmentioning

confidence: 99%

Values of inhomogeneous forms at S ‐integral points

Ghosh

Han

2022

Mathematika

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We prove effective versions of Oppenheim's conjecture for generic inhomogeneous forms in the 𝑆-arithmetic setting. We prove an effective result for fixed rational shifts and generic forms and we also prove a result where both the quadratic form and the shift are allowed to vary.In order to do so, we prove analogues of Rogers' moment formulae for 𝑆-arithmetic congruence quotients as well as for the space of affine lattices. We believe the latter results to be of independent interest.

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“…We also mention the work of Bourgain [7] on certain "uniform" versions of Oppenheim's conjecture for diagonal forms. See also [13] for an analogue for ternary forms and [5,20].…”

Section: Introductionmentioning

confidence: 99%

Values of inhomogeneous forms at S ‐integral points

Ghosh

Han

2022

Mathematika

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“…Ghosh and Kelmer [11] showed another version of the quantitative version of Oppenheim problem for real generic ternary quadratic forms and Ghosh, Gorodnik and Nevo [10] extended this result for more general setting, such as for generic characteristic polynomial maps. Recently, there have been effective results about the bound of the error terms N (a,b),Ω (T ) − V (a,b),Ω (T ) for almost all real quadratic forms of rank at least 3 by Athreya and Margulis [2], for almost all real homogeneous forms of even degree and of rank at least 3 by Kelmer and Yu [14], and for almost all systems of a homogeneous polynomial of even degree and a linear map with the certain condition by Bandi, Ghosh and the author [3].…”

Section: Introductionmentioning

confidence: 99%

“…Case i) H ∞ = SO (3,1). Set H 3 to be the 3-dimensional hyperbolic space H 3 = {z + ti : z = x + yj ∈ C and t ∈ R >0 }.…”

Section: Introductionmentioning

confidence: 99%

“…By Lemma 2.2 and Lemma 2.3, if q p is an isotropic non-split quadratic form of rank 4, then K p \ SO(q p ) is a subtree of the building B n . Lastly, if q p is a split form of rank 4, then SO(q p ) is locally isomorphic to SL 2 (Q p ) × SL 2 (Q p ) as in (3). Then K p \ SO(q p ) is properly embedded to the product T p × T p of two (p + 1)-regular trees.…”

Section: Introductionmentioning

confidence: 99%

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Quantitative oppenheim conjecture for $ S $-arithmetic quadratic forms of rank $ 3 $ and $ 4 $

Han¹

2021

Discrete &Amp; Continuous Dynamical Systems - A

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The celebrated result of Eskin, Margulis and Mozes [8] and Dani and Margulis [7] on quantitative Oppenheim conjecture says that for irrational quadratic forms q of rank at least 5, the number of integral vectors v such that q(v) is in a given bounded interval is asymptotically equal to the volume of the set of real vectors v such that q(v) is in the same interval. In rank 3 or 4, there are exceptional quadratic forms which fail to satisfy the quantitative Oppenheim conjecture. Even in those cases, one can say that two asymptotic limits coincide for almost all quadratic forms ([8, Theorem 2.4]). In this paper, we extend this result to the S-arithmetic version.

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“…We also mention the work of Bourgain [7] on certain 'uniform' versions of Oppenheim's conjecture for diagonal forms. See also [12] for an analogue for ternary forms and [4,20].…”

Section: Introductionmentioning

confidence: 99%

Values of Inhomogeneous Forms at S-integral points

Ghosh¹,

Han²

2021

Preprint

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We prove effective versions of Oppenheim's conjecture for generic inhomogeneous forms in the S-arithmetic setting. We prove an effective result for fixed rational shifts and generic forms and we also prove a result where both the quadratic form and the shift are allowed to vary. In order to do so, we prove analogues of Rogers' moment formulae for S-arithmetic congruence quotients as well as for the space of affine lattices. We believe the latter results to be of independent interest.

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A generic effective Oppenheim theorem for systems of forms

Cited by 11 publications

References 16 publications

Values of inhomogeneous forms at S ‐integral points

Values of inhomogeneous forms at S ‐integral points

Quantitative oppenheim conjecture for $ S $-arithmetic quadratic forms of rank $ 3 $ and $ 4 $

Values of Inhomogeneous Forms at S-integral points

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