The sphere problem and the L-functions

Chamizo, Fernando; Cristóbal, Elena de las Heras

doi:10.1007/s10474-011-0144-9

Cited by 10 publications

(13 citation statements)

References 21 publications

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“…The crucial step in the proof of Proposition 4.1 is a variation of the arguments of [CC12] that are a generalization of [CI95] and give a bound for…”

Section: Estimation Of the Exponential Summentioning

confidence: 99%

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Lattice points in the 3-dimensional torus

Chamizo

Raboso

2015

Journal of Mathematical Analysis and Applications

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Esta es la versión de autor del artículo publicado en: This is an author produced version of a paper published in: El acceso a la versión del editor puede requerir la suscripción del recurso Access to the published version may require subscription LATTICE POINTS IN THE 3-DIMENSIONAL TORUSFERNANDO CHAMIZO AND DULCINEA RABOSO Abstract. We prove the exponent 4/3 for the lattice point discrepancy of a torus in R 3 (generated by the rotation of a circle around the z axis). The exponent comes from a diagonal term and it seems a natural limit for any approach based solely on classical methods of exponential sums. The result extends to other solids in R 3 related to the torus.

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“…The crucial step in the proof of Proposition 4.1 is a variation of the arguments of [CC12] that are a generalization of [CI95] and give a bound for…”

Section: Estimation Of the Exponential Summentioning

confidence: 99%

“…We devote §4 to estimate some exponential sums. Part of the work is done in [CC12] and [CI95]. In §5 we combine the results of the previous sections to get Theorem 1.1.…”

Section: Introductionmentioning

confidence: 99%

Lattice points in the 3-dimensional torus

Chamizo

Raboso

2015

Journal of Mathematical Analysis and Applications

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“…In 2D, a majority of prior work focus on the characterization and generation of circles, rings, discs, and circular arcs [1,2,23,28,29,40,50,51,58,60,65,70]. In 3D, apart from straight lines and planes [14,[16][17][18]25,26,32,37,46,69], several theoretical studies on the characterization of digital spheres and hyperspheres have appeared recently [15,20,21,31,33,36,54,55,66,67].…”

Section: Introductionmentioning

confidence: 99%

“…There are some related prior work on finding the lattice points on or inside a real sphere of a given radius [15,20,21,31,36,41,55,67], and on finding a real sphere that passes through a given set of lattice points [54]. Most of them are closely related to the determination of lattice points in circles [43], ellipsoids [22,45], or surfaces of revolution [19].…”

Section: Introductionmentioning

confidence: 99%

On the Characterization of Absentee-Voxels in a Spherical Surface and Volume of Revolution in $${\mathbb Z}^3$$ Z 3

2016

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We show that the construction of a digital sphere by circularly sweeping a digital semi-circle (generatrix) around its diameter results in the appearance of some holes (absentee-voxels) in its spherical surface of revolution. This incompleteness calls for a proper characterization of the absentee-voxels whose restoration in the surface of revolution can ensure the required completeness. In this paper, we present a characterization of the absentee-voxels using certain techniques of digital geometry and show that their count varies quadratically with the radius of the semi-circular generatrix. Next, we design an algorithm to fill up the absentee-voxels so as to generate a spherical surface of revolution, which is complete and realistic from the viewpoint of visual perception. We also show how the proposed technique for absentee-filling can be used to generate a variety of digital surfaces of revolution by choosing an arbitrary curve as the generatrix. We further show that covering a solid sphere by a set of complete spheres also results to an asymptotically larger count of absentees, which is cubic in the radius of the sphere. A complete characterization of the absenteevoxels that aids the subsequent generation of a solid digital A preliminary version of this work appeared in ICAA'14 [5].

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“…Existing techniques In the literature, a multitude of works on sphere voxelation have been reported [1,2,6,7,13,30,32,39]. These are mostly of two types-one is on mathematical characterization of digital spheres (DS in short, also called discrete spheres in Z 3 [1,2,13]), and another is on voxelation.…”

Section: Introductionmentioning

confidence: 99%

Layer the sphere

Biswas

Bhowmick

2015

Vis Comput

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Voxelation today is not only limited to discretization and representation of 3D objects, but has also been gaining tremendous importance in rapid prototyping through 3D printing. In this paper, we introduce a novel technique for discretization of a sphere in the integer space, which gives rise to a set of mathematically accurate, 3D-printable physical voxels up to the desired level of precision. The proposed technique is based on an interesting correspondence between the voxel set forming a discrete sphere and certain easy-to-compute integer intervals defined by voxel position and sphere dimension. It gives us several algorithmic leverages, such as computational sufficiency with simple integer operations and amenability to layer-by-layer additive fabrication. Theoretical analysis, prototype characteristics, and experimental results demonstrate its efficiency, versatility, and further prospects.

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The sphere problem and the L-functions

Abstract: We improve the upper bound for the lattice point discrepancy of large spheres under conjectural properties of the real L-functions. In connection with this we give some new unconditional estimates for exponential and character sums of independent interest.

Cited by 10 publications

References 21 publications

Lattice points in the 3-dimensional torus

Lattice points in the 3-dimensional torus

On the Characterization of Absentee-Voxels in a Spherical Surface and Volume of Revolution in $${\mathbb Z}^3$$ Z 3

Layer the sphere

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