Translational tiles without spectra in finite abelian p-groups

Abstract

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.