On the Lattice Tilings by Lee Balls: Q-Polynomials, Group-Ring Equations, and a Decision Algorithm

Ang Xiao; Yue Zhou

Home/Archive/arXiv

arXiv · 2610.19775

On the Lattice Tilings by Lee Balls: Q-Polynomials, Group-Ring Equations, and a Decision Algorithm

Ang Xiao·Yue Zhou

View original source ↗Read source document ↗Export metadata ↓

Abstract

More than 50 years ago, Golomb and Welch conjectured that no tilings of $\mathbb{Z}^n$ by Lee balls of radius $r$ exist for $n \geq 3$ and $r \geq 2$. This conjecture remains open even under the additional assumption that the tiling is a lattice tiling. In general, there are two known effective approaches to the lattice case which are based on symmetric polynomials and on the arithmetic of group-ring equations, respectively. In this paper, we develop both approaches. First, we determine all coefficients of the Q-polynomials associated with Lee balls; previously, only leading coefficients and certain special cases were known. Second, for arbitrary radii, we give necessary and sufficient conditions for the existence of lattice tilings by Lee balls in terms of group-ring equations, extending earlier results for radii $r=2$ and $3$. We then derive new necessary conditions for such tilings. For every radius $3\le r\le 50$, we prove that the set of dimensions for which no lattice tiling exists has positive density in $\mathbb{Z}^+$. We also give an algorithm that combines the available necessary conditions and apply it to all pairs $(n,r)$ for which the corresponding Lee ball of radius $r$ and dimension $n$ has size less than $2^{30}$. Among these $1{,}358$ pairs, the combined method excludes $1{,}356$, leaving only $(n,r)=(35,3)$ and $(805,3)$.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Ang Xiao, Yue Zhou. 2026-10-14. On the Lattice Tilings by Lee Balls: Q-Polynomials, Group-Ring Equations, and a Decision Algorithm. https://arxiv.org/abs/2610.19775

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Morse theory and moduli spaces of simple polylines and polygons

We prove that a smooth $d$-manifold $M$ is diffeomorphic to $\mathbb R^d$ if it admits a Lyapunov-Reeb function, i.e., a smooth map $f\colon M\to\mathbb R$ that is proper, lower-bounded, and possesses exactly one critical point. We further establish that the moduli spaces of simple polylines and positively oriented polygons with fixed or varying side lengths are diffeomorphic to Euclidean spaces, by explicitly constructing Lyapunov-Reeb functions on these spaces. This yields a refined resolution to the Carpenter's Rule Problem and settles a conjecture proposed by González and Sedano-Mendoza. In addition, we describe foliation structures of the moduli spaces and develop algorithms for the refolding problem.

math.CO↗

arXiv2026

Linear bounds for rich-neighbor edge-colorings

An edge in a proper edge-coloring is rich if all edges adjacent to it receive pairwise distinct colors. A rich-neighbor edge-coloring is a proper edge-coloring in which every non-isolated edge is adjacent to a rich edge. The rich-neighbor chromatic index $\rn(G)$ is the minimum number of colors in such a coloring. We prove that every loopless multigraph $G$ satisfies $\rn(G)\le3χ'(G)$, where $χ'(G)$ is the chromatic index. Exact values are also determined for complete bipartite graphs and complete graphs. For all positive integers $m,n$, $\rn(K_{m,n})=m+n-1$, except that $\rn(K_{2,2})=4$. For $n\ge3$, $\rn(K_n)=2n-3$, except that $\rn(K_4)=6$ and $\rn(K_6)=10$. Moreover, for $n\ge7$, the rich edges in every optimal rich-neighbor edge-coloring of $K_n$ form a star forest.

math.CO↗

arXiv2026

Coefficient lifting for skew divided differences and triple Schubert coefficients

Skew divided difference operators $\partial_{w/v}$, introduced by Macdonald, play an important role in the study of Schubert polynomials and their structure constants. For permutations $u,v,w\in S_n$, we show that the coefficient of each monomial $\xx^α$ in $\partial_{w/v}\SSch_u(\xx;\yy)$, where $\SSch_u(\xx;\yy)$ is the double Schubert polynomial, is a single triple Schubert coefficient evaluated at $\tb=0$, whose indices form an explicit lift of $(u,v,w)$ to $S_{n+\absα}$. As an application, we show that every coefficient of $\partial_{w/v}\SSch_u(\xx)$ is a single Schubert structure constant. This gives a new proof of Kirillov's conjecture that these coefficients are nonnegative.

math.CO↗