The local contribution of zeros of curvature to lattice point asymptotics

Peter, Manfred

doi:10.1007/s002090050498

Cited by 5 publications

(3 citation statements)

References 8 publications

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“…These include the unit balls of ℓ p -norms on R n and some generalizations, and the effect of vanishing curvature on the error estimates have been investigated extensively in e.g. [Kr2], [Kr3], [Kr4], [KN1], [KN2], [No], [Pe1], [Pe2], [R1], [R2], [R3] and [R4]. It is natural to consider the following considerably more general set-up.…”

Section: Introduction Notation and Statement Of Resultsmentioning

confidence: 99%

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The lattice point counting problem on the Heisenberg groups

Garg¹,

Nevo²,

Taylor³

2015

Ann. inst. Fourier

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We consider the radial and Heisenberg-homogeneous norms on the Heisenberg groups given by N α,A ((z, t), for α ≥ 2 and A > 0. This natural family includes the canonical Cygan-Korányi norm, corresponding to α = 4. We study the lattice points counting problem on the Heisenberg groups, namely establish an error estimate for the number of points that the lattice of integral points has in a ball of large radius R. The exponent we establish for the error in the case α = 2 is the best possible, in all dimensions.

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Section: Introduction Notation and Statement Of Resultsmentioning

confidence: 99%

“…These include the unit balls of p -norms on R n and some generalizations, and the effect of vanishing curvature on the error estimates have been investigated extensively in e.g. [18], [19], [20], [21], [22], [23], [25], [26], [27], [28], [29] and [30].…”

mentioning

confidence: 99%

The lattice point counting problem on the Heisenberg groups

Garg¹,

Nevo²,

Taylor³

2015

Ann. inst. Fourier

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“…This includes error estimates in the case of super spheres [21,22] and p -balls [30,31], as well as more general surfaces of rotation [23,27]. See also [28,29] where the effect of points of vanishing curvature on the error term or "the lattice remainder term" is studied in much greater generality.…”

Section: Introductionmentioning

confidence: 99%

Lattice Points Close to the Heisenberg Spheres

Campolongo¹,

Taylor²

2022

Preprint

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We study a lattice point counting problem for spheres arising from the Heisenberg groups. In particular, we prove an upper bound on the number of points on and near large dilates of the unit spheres generated by the anisotropic norms }pz, tq} α " p|z| α `|t| α{2 q 1{α for α ě 2. As a first step, we reduce our counting problem to one of bounding an energy integral. The primary new challenges that arise are the presence of vanishing curvature and uneven dilations. In the process, we establish bounds on the Fourier transform of the surface measures arising from these norms. Further, we utilize the techniques developed here to estimate the number of lattice points in the intersection of two such surfaces.

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Precision of Geometric Moments in Picture analysis

Huxley

Klette

Žunić

Geometric Properties for Incomplete Data

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The local contribution of zeros of curvature to lattice point asymptotics

Cited by 5 publications

References 8 publications

The lattice point counting problem on the Heisenberg groups

The lattice point counting problem on the Heisenberg groups

Lattice Points Close to the Heisenberg Spheres

Precision of Geometric Moments in Picture analysis

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