The BCC lattice is the optimal 3D lattice quantizer for all $p\geq1$

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1. Apparenté Contexte plus large Pas pertinent

   A Discrete Fourier Kernel and Fraenkel's Tiling Conjecture

   Ron Graham, Kevin O'Bryant · 2004 · Acta Arith. 118 (2005), no. 3, 283--304. · cité 19 fois · Résumé IA disponible

   A Discrete Fourier Kernel étudie des partitions de Beatty et leurs propriétés de Fourier, sans traiter l’optimisation de cellules de Voronoï.

   Résumé scientifique

   The set B_{p,r}^q:=\{\floor{nq/p+r} \colon n\in Z \} with integers p, q, r) is a Beatty set with density p/q. We derive a formula for the Fourier transform \hat{B_{p,r}^q}(j):=\sum_{n=1}^p e^{-2 \pi i j \floor{nq/p+r} / q}. A. S. Fraenkel conjectured that there is essentially one way to partition the integers into m>2 Beatty sets with distinct densities. We conjecture a generalization of this, and use Fourier methods to prove several special cases of our generalized conjecture.

2. Apparenté Contexte plus large Pas pertinent

   Explicit formulas for Masses of Ternary Quadratic Lattices of varying determinant over Number Fields

   Jonathan Hanke · 2011 · Number Theory · preprint, non relu par les pairs · cité 1 fois · Résumé IA disponible

   Explicit formulas for Masses étudie des réseaux quadratiques ternaires par leurs masses arithmétiques, plutôt que par le moment de leurs cellules de Voronoï.

   Résumé scientifique

   This paper gives explicit formulas for the formal total mass Dirichlet series for integer-valued ternary quadratic lattices of varying determinant and fixed signature over number fields F where p = 2 splits completely. We prove this by using local genus invariants and local mass formulas to compute the local factors of the theory developed in [11]. When the signature is positive definite these formulas be checked against tables of positive definite ternary quadratic forms over Z, and we have written specialized software which checks these results when the Hessian determinant is \leq 2 \times 10^4. This work can also be applied to study the 2-parts of class groups of cubic fields (e.g. see [2…

3. Apparenté Contexte plus large Pas pertinent

   The State-Operator Clifford Compatibility: A Real Algebraic Framework for Quantum Information

   Kagwe A. Muchane · 2025 · Quantum Physics · Résumé IA disponible

   The State-Operator Clifford Compatibility développe un cadre algébrique pour le calcul quantique, sans étudier les quantificateurs par réseaux.

   Résumé scientifique

   We revisit the Pauli-Clifford connection to introduce a real, grade-preserving algebraic framework for $n$-qubit quantum computation based on the tensor product $C\ell_{2,0}(\mathbb{R})^{\otimes n}$. In this setting, the bivector $J = e_{12}$ satisfies $J^{2} = -1$ and supplies the complex structure on the $J$-closure of a minimal left ideal via right multiplication, while Pauli operations arise as left actions of Clifford elements. The Peirce decomposition organizes the algebra into sector blocks determined by primitive idempotents, with nilpotent elements generating transitions between sectors. Quantum states are represented as equivalence classes modulo the left annihilator, exhibiting th…

4. Apparenté Contexte plus large Pas pertinent

   Bounding the Order of Vanishing of Cuspidal Newforms via the nth Centered Moments

   Sohom Dutta, Steven J. Miller · 2022 · Number Theory · cité 3 fois · Pas encore résumé

   Bounding the Order of Vanishing utilise les moments centrés et les densités de niveaux pour les nouvelles formes cuspidales, non les moments de cellules de Voronoï.

   Résumé scientifique

   Building on the work of Iwaniec, Luo and Sarnak, we use the $n$-level density to bound the probability of vanishing to order at least $r$ at the central point for families of cuspidal newforms of prime level $N \to \infty$, split by sign. There are three methods to improve bounds on the order of vanishing: optimizing the test functions, increasing the support, and increasing the $n$-level density studied. Previous work determined the optimal test functions for the $1$ and $2$-level densities in certain support ranges, leading to marginal improvements in bounds and making it not a productive avenue for further research. Similarly the support has been increased as far as possible, and further …

5. Apparenté Contexte plus large Pas pertinent

   MINVO Basis: Finding Simplexes with Minimum Volume Enclosing Polynomial Curves

   Jesus Tordesillas, Jonathan P. How · 2020 · Computational Geometry · cité 50 fois · Résumé IA disponible

   MINVO Basis minimise le volume de simplexes contenant des courbes polynomiales, tandis que l’article minimise un moment sur des cellules de Voronoï.

   Résumé scientifique

   Outer polyhedral representations of a given polynomial curve are extensively exploited in computer graphics rendering, computer gaming, path planning for robots, and finite element simulations. Bézier curves (which use the Bernstein basis) or B-Splines are a very common choice for these polyhedral representations because their non-negativity and partition-of-unity properties guarantee that each interval of the curve is contained inside the convex hull of its control points. However, the convex hull provided by these bases is not the one with smallest volume, producing therefore undesirable levels of conservatism in all of the applications mentioned above. This paper presents the MINVO basis,…

6. Apparenté Contexte plus large Pas pertinent

   Approximating the Euclidean Ball by Polytopes: Surprising Rates in High Dimensions

   Yegor Gorodzha, Steven Hoehner, Gil Kur · 2026 · Metric Geometry · Résumé IA disponible

   Approximating the Euclidean Ball étudie l’approximation volumique d’une boule par des polytopes, et non l’optimalité des réseaux quantificateurs.

   En bref

   L’article établit les budgets de sommets nécessaires pour approximer une boule en grande dimension et montre que, à erreur fixe, ils sont sous-exponentiels, contrairement aux budgets de facettes.

7. Apparenté Contexte plus large Pas pertinent

   Potential Hessian Ascent: The Sherrington-Kirkpatrick Model

   David Jekel, Juspreet Singh Sandhu, Jonathan Shi · 2024 · Probability · preprint, non relu par les pairs · cité 8 fois · Résumé IA disponible

   Potential Hessian Ascent cherche des optima d’une fonction quadratique sur l’hypercube par un algorithme spectral, sans étudier les réseaux.

   Résumé scientifique

   We provide the first iterative spectral algorithm to find near-optima of a random quadratic objective over the discrete hypercube. The algorithm is a randomized Hessian ascent in the solid cube, where we modify the objective by subtracting a specific instance-independent potential function [Chen et al., Communications on Pure and Applied Mathematics, 76(7), 2023]. This extends Subag's algorithmic program of Hessian ascent from the sphere [Subag, Communications on Pure and Applied Mathematics, 74(5), 2021] to the more complex geometry of the cube. Utilizing tools from free probability theory, we construct an approximate projector into the top-eigenspaces of the Hessian with well-behaved diago…

8. Apparenté Contexte plus large Pas pertinent

   The clique chromatic number of sparse random graphs

   Manuel Fernandez V, Lutz Warnke · 2024 · Combinatorics · Résumé IA disponible

   The clique chromatic number détermine le nombre de couleurs des graphes aléatoires clairsemés, un problème distinct de l’optimisation des réseaux de quantification.

   Résumé scientifique

   The clique chromatic number of a graph is the smallest number of colors in a vertex coloring so that no maximal clique is monochromatic. In this paper, we determine the order of magnitude of the clique chromatic number of the random graph G_{n,p} for most edge-probabilities p in the range n^{-2/5} \ll p \ll 1. This resolves open problems and questions of Lichev, Mitsche and Warnke as well as Alon and Krievelevich. One major proof difficulty stems from high-degree vertices, which prevent maximal cliques in their neighborhoods: we deal with these vertices by an intricate union bound argument, that combines the probabilistic method with new degree counting arguments in order to enable Janson's …