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The density of the graph of elliptic Dedekind sums

Stephen Bartell, Abby Halverson, Brenden Schlader, Siena Truex, Tian An Wong

Abstract

We show that the graph of normalized elliptic Dedekind sums is dense in its image for arbitrary imaginary quadratic fields, generalizing a result of Ito in the Euclidean case. We also derive some basic properties of Martin's continued fraction algorithm for arbitrary imaginary quadratic fields.

The density of the graph of elliptic Dedekind sums

Abstract

We show that the graph of normalized elliptic Dedekind sums is dense in its image for arbitrary imaginary quadratic fields, generalizing a result of Ito in the Euclidean case. We also derive some basic properties of Martin's continued fraction algorithm for arbitrary imaginary quadratic fields.

Paper Structure

This paper contains 7 sections, 4 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. The density of elliptic Dedekind sums
  3. Main result
  4. Martin's algorithm
  5. Properties
  6. Approximation by generalized continued fractions
  7. Proof of density

Key Result

Theorem \oldthetheorem

Let $K = \mathbb{Q}(\sqrt{-D})$ with $D\neq 1,3$. Then the graph of the normalized elliptic Dedekind sum is dense in $\mathbb C\times \mathbb R$.

Theorems & Definitions (7)

  • Theorem \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof