arXiv subjects

Kirati Sriamorn

Publications and source records attributed to Kirati Sriamorn.

11 recordsLinked to original sources

arXiv2026

Multiple Lattice Packings and Coverings of the Plane with Triangles

Given a convex disk $K$ and a positive integer $j$, let $δ_L^j(K)$ and $\vartheta_L^j(K)$ denote the $j$-fold lattice packing density and the $j$-fold lattice covering density of $K$, respectively. I will prove that for every triangle $T$ we have that $δ_L^j(T)=\frac{2j^2}{2j+1}$ and $\vartheta_L^j(T)=\frac{2j+1}{2}$. Furthermore, I also obtain that the numbers of lattices which attain these densities both are $(2j+1)\prod_{p|2j+1}(1-\frac{2}{p})$, where the product is over the distinct prime numbers dividing $2j+1$.

math.MG↗

arXiv2026

Congruences via Partitions with Exactly Two Part Sizes

We prove the congruence $\sum_{1 \leq k < \sqrt{N}} σ_0 (N - k^2) \equiv 0 \pmod 4$, where $σ_0(m)$ denotes the number of positive divisors of $m$, for $N = An + B$ with $(A,B) \in \{ (16,14),$ $(36,30),$ $(72,42),$ $(196,70),$ $(252,114) \}$. Our proof relies on a result of Keith which states that $ν_2 (N) \equiv 0 \pmod 4$, where $ν_2(N)$ is the number of partitions of $N$ with exactly two part sizes. Inspired by Dewitt and Keith, our approach combines combinatorial arguments with modular arithmetic techniques.

math.NT↗

arXiv2023

Characterization of the Three-Dimensional Fivefold Translative Tiles

This paper proves the following statement: If a convex body can form a fivefold translative tiling in $\mathbb{E}^3$, it must be a parallelotope, a hexagonal prism, a rhombic dodecahedron, an elongated dodecahedron, a truncated octahedron, a cylinder over a particular octagon, or a cylinder over a particular decagon, where the octagon and the decagon are fivefold translative tiles in $\mathbb{E}^2$. Furthermore, it presents an example of multiple tiles in $\mathbb{E}^3$ with multiplicity at most 10 which is neither a parallelohedron nor a cylinder.

math.MG↗

arXiv2023

On the multiple illumination numbers of convex bodies

In this paper, we introduce an $m$-fold illumination number $I^m(K)$ of a convex body $K$ in Euclidean space $\mathbb{E}^d$, which is the smallest number of directions required to $m$-fold illuminate $K$, i.e., each point on the boundary of $K$ is illuminated by at least $m$ directions. We get a lower bound of $I^m(K)$ for any $d$-dimensional convex body $K$, and get an upper bound of $I^m(\mathbb{B}^d)$, where $\mathbb{B}^d$ is a $d$-dimensional unit ball. We also prove that $I^m(K)=2m+1$, for a $2$-dimensional smooth convex body $K$. Furthermore, we obtain some results related to the $m$-fold illumination numbers of convex polygons and cap bodies of $\mathbb{B}^d$ in small dimensions. In particular, we show that $I^m(P)=\left\lceil mn/{\left\lfloor\frac{n-1}{2}\right\rfloor}\right\rceil$, for a regular convex $n$-sided polygon $P$.

math.MG↗

arXiv2021

The Three and Fourfold Translative Tiles in Three-Dimensional Space

This paper proves the following statement: If a convex body can form a three or fourfold translative tiling in the three-dimensional space, it must be a parallelohedron. In other words, it must be a parallelotope, a hexagonal prism, a rhombic dodecahedron, an elongated dodecahedron, or a truncated octahedron.

math.MG↗

arXiv2021

Twofold Translative Tiles in Three-Dimensional Space

This paper proves the following statement: {\it If a convex body can form a twofold translative tiling in $\mathbb{E}^3$, it must be a parallelohedron.} In other words, it must be a parallelotope, a hexagonal prism, a rhombic dodecahedron, an elongated dodecahedron, or a truncated octahedron.

math.MG↗

arXiv2016

On the Covering Densities of Quarter-Convex Disks

It is conjectured that for every convex disks K, the translative covering density of K and the lattice covering density of K are identical. It is well known that this conjecture is true for every centrally symmetric convex disks. For the non-symmetric case, we only know that the conjecture is true for triangles. In this paper, we prove the conjecture for a class of convex disks (quarter-convex disks), which includes all triangles and convex quadrilaterals.

math.MG↗

arXiv2016

On the Multiple Covering Densities of Triangles

Given a convex disk $K$ and a positive integer $k$, let $\vartheta_T^k(K)$ and $\vartheta_L^k(K)$ denote the $k$-fold translative covering density and the $k$-fold lattice covering density of $K$, respectively. Let $T$ be a triangle. In a very recent paper, K. Sriamorn proved that $\vartheta_L^k(T)=\frac{2k+1}{2}$. In this paper, we will show that $\vartheta_T^k(T)=\vartheta_L^k(T)$.

math.MG↗

arXiv2016

On the Multiple Packing Densities of Triangles

Given a convex disk $K$ and a positive integer $k$, let $δ_T^k(K)$ and $δ_L^k(K)$ denote the $k$-fold translative packing density and the $k$-fold lattice packing density of $K$, respectively. Let $T$ be a triangle. In a very recent paper, K. Sriamorn proved that $δ_L^k(T)=\frac{2k^2}{2k+1}$. In this paper, I will show that $δ_T^k(T)=δ_L^k(T)$.

math.MG↗

arXiv2016

Twofold Translative Tilings with Convex Bodies

Let $K$ be a convex body. It is known that, in general, if $K$ is a $k$-fold translative tile (for some positive integer $k$), then $K$ may not be a (onefold) translative tile. However, in this paper I will show that for every convex body $K$, $K$ is a twofold translative tile if and only if $K$ is a translative tile.

math.MG↗

arXiv2014

On the Lattice Packings and Coverings of the Plane with Convex Quadrilaterals

It is well known that the lattice packing density and the lattice covering density of a triangle are $\frac{2}{3}$ and $\frac{3}{2}$ respectively. We also know that the lattices that attain these densities both are unique. Let $δ_{L}(K)$ and $\vartheta_{L}(K)$ denote the lattice packing density and the lattice covering density of $K$, respectively. In this paper, I study the lattice packings and coverings for a special class of convex disks, which includes all triangles and convex quadrilaterals. In particular, I determine the densities $δ_{L}(Q)$ and $\vartheta_{L}(Q)$, where $Q$ is an arbitrary convex quadrilateral. Furthermore, I also obtain all of lattices that attain these densities. Finally, I show that $δ_{L}(Q)\vartheta_{L}(Q)\geq 1$ and $\frac{1}{δ_{L}(Q)}+\frac{1}{\vartheta_{L}(Q)}\geq 2$, for each convex quadrilateral $Q$.

math.MG↗