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Classification of rational angles in plane lattices

Roberto Dvornicich, Francesco Veneziano, Umberto Zannier

Abstract

This paper is concerned with configurations of points in a plane lattice which determine angles that are rational multiples of $π$. We shall study how many such angles may appear in a given lattice and in which positions, allowing the lattice to vary arbitrarily. This classification turns out to be much less simple than could be expected, leading even to parametrizations involving rational points on certain algebraic curves of positive genus.Bulletin of the American Mathematical Society

Classification of rational angles in plane lattices

Abstract

This paper is concerned with configurations of points in a plane lattice which determine angles that are rational multiples of . We shall study how many such angles may appear in a given lattice and in which positions, allowing the lattice to vary arbitrarily. This classification turns out to be much less simple than could be expected, leading even to parametrizations involving rational points on certain algebraic curves of positive genus.Bulletin of the American Mathematical Society

Paper Structure

This paper contains 52 sections, 13 theorems, 104 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Rational angles in the Gaussian lattice.
  3. On Sherrer's proof.
  4. Terminology and notation
  5. Equations
  6. The equation of a rational angle
  7. The equations of a rational n-tuple containing 1
  8. The equations of a rational n-tuple not containing 1
  9. Equations for the three main cases
  10. Equations in roots of unity
  11. Spaces with special symmetries
  12. Spaces with V=V conj
  13. CM spaces
  14. Spaces with V hom V conj
  15. Finiteness of angles in lattices outside special families
  16. ...and 37 more sections

Key Result

Theorem 1.1

Let $V$ be a space of one of the three types \begin{tikzpicture}[baseline=(char.base)]{ \node[shape=circle,draw,inner sep=2pt] (char) {2};}\end{tikzpicture}+\begin{tikzpicture}[baseline=(char.base)]{ \node[shape=circle,draw,inner sep=2pt] (char) {2};}\end{tikzpicture}+\begin{

Figures (3)

  • Figure 1:
  • Figure 2:
  • Figure 3:

Theorems & Definitions (28)

  • Theorem 1.1
  • Remark 2.1
  • Remark 2.2
  • Remark 3.1
  • Theorem 3.2: Conway-Jones
  • Lemma 4.1
  • proof
  • Theorem 4.2
  • proof
  • Theorem 4.3
  • ...and 18 more