Mean square value of exponential sums related to the representation of integers as sums of squares

Let G = SL(2, R) ⋉ (R 2 ) ⊕k and let Γ be a congruence subgroup of SL(2, Z) ⋉ (Z 2 ) ⊕k . We prove a polynomially effective asymptotic equidistribution result for special types of unipotent orbits in Γ\G which project to pieces of closed horocycles in SL(2, Z)\ SL(2, R).As an application, we prove an effective quantitative Oppenheim type result for the quadraticfollowing the approach by Marklof [24] using theta sums.1 Apply [5, Thm. 3] with d = 2 and M = 12 and use the anti-automorphism (M, ( x x ′ )) → ( t M, t M x ′ −x ) of G to translate from the setting with G/Γ in [5] into our setting with X = Γ\G. As noted in [5, Remark 7.2], the proof of [5, Thm. 3] extends trivially to the case when Γ is an arbitrary subgroup of SL(2, Z) ⋉ (Z 2 ) ⊕k of finite index.

show abstract

“…The estimate of the form R α (N ) N d/2+ was announced in [1] under a stronger condition that α should be multiplicatively Diophantine. We refer to [12] for further comparison and more detailed discussion.…”

mentioning

confidence: 99%

“…Properties of r α (n). The proof of Theorem 2.2 relies on the following delicate asymptotic result for R α (N ) (see (7)), established in [12]. PROPOSITION 2.3.…”

mentioning

confidence: 99%

Distribution of Integer Lattice Points in a Ball Centred at a Diophantine Point

Kang

Sobolev

2009

Mathematika

View full text Add to dashboard Cite

Abstract. We study the variance of the fluctuations in the number of lattice points in a ball and in a thin spherical shell of large radius centred at a Diophantine point. §1. Introduction. The distribution of lattice points has been extensively studied in the literature for its own sake, as well as with the aim of understanding the clustering of eigenvalues of quantum Hamiltonians associated with integrable systems. The eigenvalues of the "shifted" Laplaciangiven by the numbers |m − α| 2 , m ∈ Z d , and hence their counting function coincides with the number N (t) of lattice points inside a ball of a radius t, centred at α. It is immediately seen thatwhere B d is the volume of the unit ball in R d . Our object is the distribution of N (t), as a function of large t for a fixed α, in two regimes. First, we studyi.e. the normalized deviation of N (t) from its asymptotic value. Secondly, for ρ ∈ (0, 1), we investigatewhich is the normalized deviation of the number of lattice points in the spherical shell between the spheres of radii t + ρ and t from its asymptotics. Our aim is to study the asymptotics of weighted averages of F and S as t → ∞ and, in the case of S, as ρ → 0. Introduce a non-negative function ω ∈ C ∞ 0 (R) such that ω(t) = 0 for all t ≤ t 0 with some t 0 > 0 and ω(t) dt = 1. With the smooth measure induced by ω, we define for all T > 0 an averaging operator for a function f ∈ L 1 loc (R) by

show abstract

Mean square value of exponential sums related to the representation of integers as sums of squares

Cited by 10 publications

References 12 publications

An effective Ratner equidistribution result for SL(2,R)⋉R2

An effective Ratner equidistribution result for SL(2,R)⋉R2

An effective equidistribution result for SL(2,R)⋉(R2)⊕k and application to inhomogeneous quadratic forms

Distribution of Integer Lattice Points in a Ball Centred at a Diophantine Point

Contact Info

Product

Resources

About