Equidistribution of values of linear forms on quadratic surfaces

Sargent, Oliver

doi:10.2140/ant.2014.8.895

Cited by 11 publications

(10 citation statements)

References 14 publications

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“…Proof. This proposition is analogous to Lemma 3.6 in [13], and a special case of Lemma 5.1 in [28] where the number of linear forms is set to be one, and the matrix g to the identity. Let us point out that the value of the quadratic form denoted by a is assumed to be fixed and is not included as one of the arguments of J f .…”

Section: Passage To Dynamics On the Space X N Of Unimodular Latticesmentioning

confidence: 83%

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Asymptotic distribution of values of isotropic here quadratic forms at <i>S</i>-integral points

Han¹,

Lim²,

Mallahi-Karai³

2017

Journal of Modern Dynamics

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We prove an analogue of a theorem of Eskin-Margulis-Mozes [10]: suppose we are given a finite set of places S over Q containing the archimedean place and excluding the prime 2, an irrational isotropic form q of rank n ≥ 4 on QS, a product of p-adic intervals Ip, and a product Ω of star-shaped sets.We show that unless n = 4 and q is split in at least one place, the number of S-integral vectors v ∈ TΩ satisfying simultaneously q(v) ∈ Ip for p ∈ S is asymptotically given by λ(q, Ω)|I| · T n−2 ,as T goes to infinity, where |I| is the product of Haar measures of the p-adic intervals Ip.The proof uses dynamics of unipotent flows on S-arithmetic homogeneous spaces; in particular, it relies on an equidistribution result for certain translates of orbits applied to test functions with a controlled growth at infinity, specified by an S-arithmetic variant of the α-function introduced in [10], and an S-arithemtic version of a theorem of Dani-Margulis [7].The proof of Theorem 1.5 rests upon a number of ingredients. We will use the dynamics on G/Γ, where G = SL n (Q S ) and Γ = SL n (Z S ), which can be identified with a set L S of unimodular S-lattices. First, we will relate the counting problem to a question about the asymptotic behavior of integrals of the form K f (a t k∆)dm(k), where K is a maximal compact subgroup of SO(q), x is an element in G/Γ related to q, m is the normalized Haar measure of K, and a t is a 1-parameter diagonal subgroup of SO(q) (see Equation (3) in Section 2.3). Here, f is the Siegel transform of a compactly supported function f defined on Q n S (see Definition 3.7). Such integrals, when f is replaced by a bounded continuous function can be dealt with using an S-arithmetic version of the results in [7], which we will state and prove in Section 6 of this paper.Since f is unbounded, in order to use the equidistribution result just described, one needs to also control the integral of f (a t k∆) when a t k∆ is far into the cusp. As in [10], this is dealt with using the function α S (see section 3.6), which is an analog of α introduced in the previous subsection. We prove Theorem 1.7. With the notation as above, assume that q is not exceptional. For 0 < s < 2 and for any S-lattice ∆ ∈ L S , sup t≻1 K α S (a t k∆) s dm(k) < ∞.Moreover, the bound is uniform as ∆ varies over a compact subset C of L S .

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Section: Passage To Dynamics On the Space X N Of Unimodular Latticesmentioning

confidence: 83%

“…For each λ, denote by V λ the direct sum of all irreducible sub-representations of ρ with highest weight λ. Denote by τ λ the canonical orthogonal projection of i (R n ) onto V λ . Fix ε > 0 and 0 < i < n. Following [6], [28] define Benoist-Quint ϕ-function…”

Section: Strategy Of Proofmentioning

confidence: 99%

Asymptotic distribution of values of isotropic here quadratic forms at <i>S</i>-integral points

Han¹,

Lim²,

Mallahi-Karai³

2017

Journal of Modern Dynamics

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“…Sargent computed the density of M ({v ∈ Z n : q(v) = a}) ⊆ R, where q is a rational quadratic form and M is a linear map [23]. In [24], he showed the quantitative version of his result.…”

Section: Introductionmentioning

confidence: 99%

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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“…However, a similar problem, that of the effective density of linear maps taking values on rational quadratic surfaces can be addressed using ergodic methods, see Theorem 1.5 in [11]. Previously density and counting results in this setting were proved by Sargent [25,26]. 1.2.…”

Section: Introductionmentioning

confidence: 99%