发表机构
School of Mathematics and Statistics, Central China Normal University; School of Mathematics and Statistics, Kashi University（华中师范大学数学与统计学院; 喀什大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究在三类有限阿贝尔p-群中构造出无谱平移tile，证明有限阿贝尔p-群中tile到谱的蕴含关系不成立，还给出两种构造机制并通过程序验证相关断言。
AI 中文摘要
我们在三个有限阿贝尔p-群中构造了明确的无谱平移tile：第一个是ℤ₄⁴×ℤ₂²的64点子集，另外两个分别是𝔽₂¹³的512点子集和𝔽₃⁹的2187点子集。因此，有限阿贝尔p-群中tile到谱的蕴含关系不成立，且对p=2和p=3，该关系在初等阿贝尔群类中已不成立。两个基本机制构成这些例子：两层阻碍将带有两个合适 tiling 补集的谱非tile转化为无谱tile；纤维-团阻碍将带有受控公共傅里叶零点的 tiling 补集族转化为初等阿贝尔反例。所有坐标数据均已包含，有限断言由三个简短自包含程序通过精确整数运算和穷举搜索验证，附带的源文件可重新计算证明中用到的所有断言。
英文摘要
We construct explicit translational tiles without spectra in three finite abelian $p$-groups. The first is a $64$-point subset of $\Z_4^4\times\Z_2^2$. The other two are a $512$-point subset of $\F_2^{13}$ and a $2187$-point subset of $\F_3^9$. Consequently, the tile-to-spectral implication fails for finite abelian $p$-groups, and it already fails within the class of elementary abelian groups for both $p=2$ and $p=3$. Two elementary mechanisms organize the examples. A two-layer obstruction turns a spectral non-tile with two suitable tiling complements into a tile without a spectrum. A fiber--clique obstruction converts a family of tiling complements with controlled common Fourier zeros into an elementary abelian counterexample. All coordinate data are included. The finite claims are certified by three short, self-contained programs using exact integer arithmetic and exhaustive searches; the accompanying source files recompute every assertion used in the proofs.
Comments10 pages
