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On the arithmetic and algebraic properties of Minkowski balls and spheres

Nikolaj Glazunov

TL;DR

This paper surveys and develops the arithmetic and algebraic properties of Minkowski balls $D_p$ and Minkowski spheres ${\mathbb S}^{n-1}_c$, linking lattice geometry, number theory, matroid theory, and absolute algebraic geometry. It advances packing and covering theory in the plane for Minkowski bodies, introduces a moduli space of inscribed hexagons, and establishes dualities between lattices and coverings, with explicit bounds and numerical exemplars (e.g., $\vartheta(D_3)\approx1.0567$). In parallel, the work develops average-value phenomena for number-theoretic objects on arithmetic spheres, proving maximal inequalities, ergodic theorems, equidistribution, and discrepancy estimates. It further expands into direct systems, dualities, association schemes, spherical designs, and algebro-geometric aspects such as Riemann–Roch for lattices and arithmetic curves, together with algebraic dynamics and non-Archimedean Minkowski curves. Finally, the paper explores matroid theory, including metrizational frameworks, and connects matroids with Minkowski spheres, highlighting structural and polynomial invariants relevant to these geometric constructs.

Abstract

This paper gives a brief overview of some new work in number theory and algebra, and also studies the arithmetic and algebraic properties of Minkowski balls and spheres. The content of the paper is presented in more detail in the table of Contents and in the Introduction.

On the arithmetic and algebraic properties of Minkowski balls and spheres

TL;DR

This paper surveys and develops the arithmetic and algebraic properties of Minkowski balls and Minkowski spheres , linking lattice geometry, number theory, matroid theory, and absolute algebraic geometry. It advances packing and covering theory in the plane for Minkowski bodies, introduces a moduli space of inscribed hexagons, and establishes dualities between lattices and coverings, with explicit bounds and numerical exemplars (e.g., ). In parallel, the work develops average-value phenomena for number-theoretic objects on arithmetic spheres, proving maximal inequalities, ergodic theorems, equidistribution, and discrepancy estimates. It further expands into direct systems, dualities, association schemes, spherical designs, and algebro-geometric aspects such as Riemann–Roch for lattices and arithmetic curves, together with algebraic dynamics and non-Archimedean Minkowski curves. Finally, the paper explores matroid theory, including metrizational frameworks, and connects matroids with Minkowski spheres, highlighting structural and polynomial invariants relevant to these geometric constructs.

Abstract

This paper gives a brief overview of some new work in number theory and algebra, and also studies the arithmetic and algebraic properties of Minkowski balls and spheres. The content of the paper is presented in more detail in the table of Contents and in the Introduction.
Paper Structure (55 sections, 28 theorems, 52 equations)

This paper contains 55 sections, 28 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Minkowski balls and spheres
  3. About the average values of number-theoretic objects
  4. Packing of Minkowski balls
  5. Minkowski balls and the density of the packing of $2$-dimensional Minkowski balls
  6. On coverings by Minkowski balls
  7. Two-dimensional Minkowski balls and hexagons.
  8. Hexagons and coverings.
  9. Inscribed hexagons of admissible lattices of Minkowski balls $D_p$ and their moduli space.
  10. Sections of moduli space of hexagons.
  11. Estimates and bounds on covering constants.
  12. Estimates and calculation of covering density
  13. Critical lattices, Minkowski domains and their optimal packing
  14. Domains.
  15. Inscribed and circumscribed hexagons of minimum areas
  16. ...and 40 more sections

Key Result

Proposition 1

Casselslek. A lattice $\Lambda$ is a packing lattice of $D_p$ if and only if it is admissible lattice for $2 D_p$.

Theorems & Definitions (66)

  • Remark 1
  • Remark 2
  • Remark
  • Remark 3
  • Definition 1
  • Definition 2
  • Proposition 1
  • Proposition 2
  • Theorem 1
  • Remark 4
  • ...and 56 more