Interior Hulls of Clean Lattice Parallelograms and Continued Fractions
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arXiv
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2024
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| author | Khan, Gabriel Khan, Mizan R. Khan, Riaz R. Zhao, Peng |
| author_facet | Khan, Gabriel Khan, Mizan R. Khan, Riaz R. Zhao, Peng |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_04356 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Interior Hulls of Clean Lattice Parallelograms and Continued Fractions Khan, Gabriel Khan, Mizan R. Khan, Riaz R. Zhao, Peng Number Theory 52C05, 11A55, 11A05 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. |
| title | Interior Hulls of Clean Lattice Parallelograms and Continued Fractions |
| topic | Number Theory 52C05, 11A55, 11A05 |
| url | https://arxiv.org/abs/2401.04356 |