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Interior Hulls of Clean Lattice Parallelograms and Continued Fractions

Gabriel Khan, Mizan R. Khan, Riaz R. Khan, Peng Zhao

Abstract

The interior hull of a lattice polygon is the convex closure of the lattice points in the interior of the polygon. In this paper we give a concrete description of the interior hull of a clean lattice parallelogram. A clean parallelogram in $\mathbb{R}^2$ is a lattice parallelogram whose boundary contains no lattice points other than its vertices. Using unimodular maps we can identify a clean parallelogram with a parallelogram, $P_{a,n}$, whose vertices are $(0,0), (1,0), (a,n)$ and $(a+1,n)$, with $0<a <n$ and $\gcd(a,n)=1$. Following Stark's geometric approach to continued fractions we show that the convergents of the continued fraction of $n/a$ (viewed as lattice points) appear in a one-to-two correspondence with the vertices of the interior hull of this parallelogram. Consequently, if the continued fraction of $n/a$ has many partial quotients, then the interior hull of the corresponding parallelogram has many vertices. A pleasing consequence of our work is that we obtain an elementary geometric interpretation of the sum of the partial quotients of the continued fraction of $n/a$. Specifically, it is the difference between the area of the clean parallelogram $P_{a,n}$ and the area of its interior hull.

Interior Hulls of Clean Lattice Parallelograms and Continued Fractions

Abstract

The interior hull of a lattice polygon is the convex closure of the lattice points in the interior of the polygon. In this paper we give a concrete description of the interior hull of a clean lattice parallelogram. A clean parallelogram in is a lattice parallelogram whose boundary contains no lattice points other than its vertices. Using unimodular maps we can identify a clean parallelogram with a parallelogram, , whose vertices are and , with and . Following Stark's geometric approach to continued fractions we show that the convergents of the continued fraction of (viewed as lattice points) appear in a one-to-two correspondence with the vertices of the interior hull of this parallelogram. Consequently, if the continued fraction of has many partial quotients, then the interior hull of the corresponding parallelogram has many vertices. A pleasing consequence of our work is that we obtain an elementary geometric interpretation of the sum of the partial quotients of the continued fraction of . Specifically, it is the difference between the area of the clean parallelogram and the area of its interior hull.
Paper Structure (14 sections, 20 theorems, 78 equations, 4 figures)

This paper contains 14 sections, 20 theorems, 78 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Clean Parallelograms and a Reduction Result
  4. Stark's geometric exposition of the continued fraction algorithm
  5. Continued fractions and the boundary of $P_{a,n}^{(1)}$
  6. A menagerie of notations and preliminary results
  7. Our main theorem
  8. Some numeric and geometric consequences
  9. A couple of averages
  10. Speculative connections to modular forms?
  11. Exercise for the curious reader.
  12. Some future directions.
  13. An interesting degenerate case in $\mathbb R^3$.
  14. Appendix: Graphing Routines

Key Result

Theorem 4

Let $\mathbf{u}=(u_1,u_2),\mathbf{v}=(v_1,v_2) \in \mathbb Z^2$ be linearly independent. Then, That is, the cardinality of the quotient group equals the area of the parallelogram spanned by $\mathbf{u},\mathbf{v}$.

Figures (4)

  • Figure 1: The CF algorithm
  • Figure 2: Subdividing $P_{a,n}$
  • Figure 3: A schematic diagram of $P_{a,n}$ with $R_0 =P_{a,n}^{(1)}$
  • Figure 4: $P_{11,29}$ and $P_{11,29}^{(1)}$

Theorems & Definitions (32)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 4
  • Theorem 5: Pick
  • Lemma 6
  • Definition 7
  • Theorem 8
  • proof
  • Lemma 9
  • ...and 22 more