Mutual Annihilation of Tiles

We prove that if the zero set of the Fourier transform of $A\subseteq\mathbb Z_n\times\mathbb Z_n$ contains an element of prime power order, then there is an equi-distribution relation in subsets of $A$ with respect to certain hyperplanes. With this we further show that if $A$ is a tiling complement of the subgroup generated by $(p,0)$ and $(0,p)$ in $\mathbb Z_{p^m}\times\mathbb Z_{p^m}$, then the zero set of its Fourier transform is disjoint with the orthogonal rotation of $A$. These results are motivated by a casual observation in $\mathbb Z_{p^2}\times\mathbb Z_{p^2}$.

💡 Research Summary

The paper investigates the interplay between tiling sets and spectral sets in the finite abelian group (\mathbb Z_n\times\mathbb Z_n). The author focuses on the situation where the Fourier transform of the indicator function of a set (A\subseteq\mathbb Z_n^2) possesses a zero at a group element whose order is a prime power. The main result, Theorem 1, shows that such a zero forces a very rigid equi‑distribution of the points of (A) with respect to a family of affine hyperplanes. More precisely, if (x\in\mathbb Z_n^2) has order (p^m) (with (p) prime) and (\widehat{1_A}(x)=0), then (i) the number of points of (A) lying in the orthogonal complement of the cyclic subgroup generated by (x) is multiplied by (p) when the subgroup is replaced by its (p)-multiple, i.e. (|A\cap\langle px\rangle^\perp|=p,|A\cap\langle x\rangle^\perp|); and (ii) the cardinalities of the intersections (A\cap V(k)_x) are all equal for the (p) distinct residue classes (k\in{0,n/p,\dots,n(p-1)/p}), where (V(k)x={y\in\mathbb Z_n^2:\langle y,x\rangle\equiv k\pmod n}). The proof rewrites the Fourier sum (\widehat{1_A}(x)=\sum{a\in A}e^{2\pi i\langle a,x\rangle/n}) as a polynomial in (\omega=e^{2\pi i/p^m}) and uses the fact that a zero forces divisibility by the (p^m)-th cyclotomic polynomial. By expanding the product one sees that each coefficient counts points of (A) in a specific residue class, which yields the equalities above.

Armed with this structural information, the author turns to a concrete subgroup (K) of (\mathbb Z_{p^m}^2) generated by ((p,0)) and ((0,p)). This subgroup is non‑cyclic and has order (p^{2(m-1)}). A set (A) is called a tiling complement of (K) if (A\oplus K=\mathbb Z_{p^m}^2); equivalently (|A|=p^{2(m-1)}) and the difference sets (\Delta A) and (\Delta K) are disjoint. Lemma 1 proves that for any such tiling complement (A) the orthogonal rotation (A^\perp={(-a_2,a_1):a=(a_1,a_2)\in A}) contains no Fourier zero of (A). The argument proceeds by contradiction: assuming some non‑zero (a^\perp) lies in the zero set, Theorem 1 forces a relation between the sizes of (A\cap\langle a^\perp\rangle^\perp) and (A\cap\langle pa^\perp\rangle^\perp). Because (A) tiles with (K), the latter intersection can have at most (p) elements, which together with the forced equality would imply (|A\cap\langle a^\perp\rangle^\perp|\le1). Yet the subgroup (\langle a^\perp\rangle^\perp) always contains at least the origin and the original point (a), a contradiction. Hence no such zero can exist.

Corollary 1 extends this disjointness from the rotated set itself to its difference set: (\Delta(A^\perp)\cap Z(\widehat{1_A})=\varnothing). If a difference (a^\perp-b^\perp) were a Fourier zero, translating (A) by (-b) yields another tiling complement (B=A-b) whose rotated set still contains the same zero, contradicting Lemma 1. Consequently the difference set of the rotated tile is also free of Fourier zeros. In the special case (m=2) this reduces to the statement (\Delta A\cap Z(\widehat{1_A})=\varnothing), exactly the phenomenon observed in the introductory example.

The paper notes that the disjointness property holds more generally for any subgroup (H) because the Fourier transform of a subgroup is a scaled indicator of its orthogonal complement, i.e. (\widehat{1_H}=|H|,1_{H^\perp}). Thus (\Delta(H^\perp)\cap Z(\widehat{1_H})=\varnothing) for all subgroups, and the same argument applies to tiling complements of proper subgroups in (\mathbb Z_p^2).

Overall, the work provides a clear link between the existence of prime‑power order zeros of a Fourier transform and an equi‑distribution phenomenon on affine hyperplanes. It then leverages this link to prove that for a natural class of non‑cyclic subgroups in (\mathbb Z_{p^m}^2) the tiling complement’s Fourier zeros avoid the orthogonal rotation of the tile, and even avoid the differences of that rotation. These results contribute to the broader program of understanding when tiling sets are also spectral, and suggest that in certain structured settings a “universal spectrum” may exist for all tiling complements of a given subgroup.