On the density at integer points of a system comprising an inhomogeneous quadratic form and a linear form

Bandi, Prasuna; Ghosh, Anish

doi:10.1007/s00209-021-02716-8

Cited by 5 publications

(3 citation statements)

References 22 publications

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“…It also generalizes results of Sarnak [Sar97] (for the homogeneous case), and Marklof [Mar02,Mar03] (for the inhomogeneous case) who considered an important special case related to the pair correlation of values of a positive definite form. In particular, one can deduce from this an analogue of the famous Oppenheim conjecture for inhomogeneous forms, stating that for any indefinite, irrational, non-degenerate inhomogeneous form Q α in n ≥ 3 variables Q α (Z n ) is dense in R (see also [BG19] for a self-contained proof).…”

Section: Introductionmentioning

confidence: 97%

Effective density for inhomogeneous quadratic forms I: generic forms and fixed shifts

Ghosh,

Kelmer,

2019

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We establish effective versions of Oppenheim's conjecture for generic inhomogeneous quadratic forms. We prove such results for fixed shift vectors and generic quadratic forms. When the shift is rational we prove a counting result which implies the optimal density for values of generic inhomogeneous forms. We also obtain a similar density result for fixed irrational shifts satisfying an explicit Diophantine condition. The main technical tool is a formula for the second moment of Siegel transforms on certain congruence quotients of SL n (R) which we believe to be of independent interest. In a sequel, we use different techniques to treat the companion problem concerning generic shifts and fixed quadratic forms.

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Section: Introductionmentioning

confidence: 97%

Effective density for inhomogeneous quadratic forms I: generic forms and fixed shifts

Ghosh,

Kelmer,

2019

Preprint

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“…Our general framework also allows for easy consideration of vector-valued examples of f. For instance, let f 1 = F p,q,β : R n → R as in Remark 1.10, and let f 2 : R n → R be an R-linear transformation. (These functions may remind the reader of the setting in the papers [3,6,12].) Then f := (f 1 , f 2 ) : R n → R 2 satisfies the hypotheses of Theorem 1.7 if and only if the intersection Z(f 1 ) ∩ Z(f 2 ) is nonempty and transverse.…”

Section: Introductionmentioning

confidence: 99%

Inhomogeneous Diophantine approximation for generic homogeneous functions

Kleinbock¹,

Skenderi²

2022

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The present paper is a sequel to [Monatsh. Math. 194 (2021), 523-554] in which results of that paper are generalized so that they hold in the setting of inhomogeneous Diophantine approximation. Given any integers n ≥ 2 and ℓ ≥ 1, any ξ ξ ξ = (ξ 1 , . . . , ξ ℓ ) ∈ R ℓ , and any homogeneous function f = (f 1 , . . . , f ℓ ) : R n → R ℓ that satisfies a certain nonsingularity assumption, we obtain a biconditional criterion on the approximating function ψ = (ψ 1 , . . . , ψ ℓ ) : R ≥0 → (R >0 ) ℓ for a generic element in the G-orbit of f to be (respectively, not to be) ψ-approximable at ξ ξ ξ: that is, for there to exist infinitely many (respectively, only finitely many) v ∈ Z n such that |ξ j − (f j • g) (v)| ≤ ψ j ( v ) for each j ∈ {1, . . . , ℓ}. In this setting, we also obtain a sufficient condition for uniform approximation. We also consider some examples of f that do not satisfy our nonsingularity assumptions and prove similar results for these examples. Here, G can be any closed subgroup of ASLn(R) (such as ASLn(R) itself or SLn(R)) that satisfies certain axioms introduced by the authors in the aforementioned previous paper.

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“…Then the inhomogeneous quadratic form is F (x + θ) for x ∈ R s . Values of inhomogeneous forms at integer points have been studied extensively [4,[25][26][27]33]. In [15,16], Ghosh, Kelmer and Yu proved effective results for inhomogeneous quadratic forms.…”

Section: Introductionmentioning

confidence: 99%