Fuglede's Conjecture on Cyclic Groups of Square-Free Order: The Case of Rapidly Growing Prime Factors

Technical Summary: Fuglede's Conjecture on Cyclic Groups of Square-Free Order

Problem Statement
The paper addresses Fuglede's conjecture in the context of finite cyclic groups. The conjecture posits that a set is spectral (admits an orthogonal basis of exponential functions) if and only if it tiles the space by translation. While the conjecture has been disproven in Euclidean spaces of dimension $d \ge 2$, the one-dimensional case remains open. Through reductions established by Dutkay and Lai, and recent work by Fu and Song proving the rationality of normalized spectra, the conjecture on the real line $\mathbb{R}$ is equivalent to the conjecture holding for all finite cyclic groups $\mathbb{Z}_N$.

Despite substantial progress, the conjecture remains open for general finite cyclic groups. Specifically, while the "tile-to-spectral" direction (T-S) is known to hold for all square-free order cyclic groups, the "spectral-to-tiling" direction (S-T) has only been verified for specific families (e.g., prime powers, groups with few prime factors, or specific large-prime configurations). A critical gap exists: prior to this work, no infinite family of cyclic groups was known to satisfy the conjecture that possessed an arbitrary number of distinct prime factors.

Methodology
The author employs an inductive technique rooted in cyclotomic divisibility and Fourier analysis. The core methodology involves:

1. Cyclotomic Formulation: Utilizing mask polynomials $A(X)$ and the condition that $\Phi_m(X)$ divides $A(X)$ to characterize spectrality.
2. Cube Rule and Fiber Decomposition: Applying the "cube rule" (Proposition 2.1) and levelwise versions (Lemma 2.2) to analyze the structure of spectral sets within product groups. This involves decomposing a set $A \subset \mathbb{Z}_n \times \mathbb{Z}_p$ into "levels" (cosets of $\mathbb{Z}_n$) and analyzing their properties.
3. Inductive Step: The paper establishes a stability result: if the S-T direction holds for a square-free group $\mathbb{Z}_n$, it also holds for $\mathbb{Z}_n \times \mathbb{Z}_p$ provided $p > n$ is prime.
   • Case 1 ($p \nmid |A|$): The author proves that multiplication by $p$ is injective on the spectral set $A$. This allows projecting $A$ to a spectral set in $\mathbb{Z}_n$, where the inductive hypothesis applies.
   • Case 2 ($p \mid |A|$): The author proves that if $A$ is spectral in $\mathbb{Z}_n \times \mathbb{Z}_p$ and $p$ divides its cardinality, then all "levels" of $A$ (intersections with cosets of $\mathbb{Z}_n$) share a common spectrum $E \subset \mathbb{Z}_n$ (Proposition 3.2).
4. Coven-Meyerowitz Connection: In the case where $n$ is square-free, the Coven-Meyerowitz conjecture (specifically the result by Laba and Meyerowitz) guarantees that any tile of a given cardinality has a standard subgroup complement. This ensures that the common spectrum $E$ found in the levels implies that all levels tile $\mathbb{Z}_n$ with the same complement, allowing the construction of a global tiling for $A$.

Key Contributions and Results
The primary contribution is the establishment of an inductive argument that extends the validity of Fuglede's conjecture to new infinite families of cyclic groups.

• Theorem 1.1 (Main Result): Let $n$ be square-free and $p > n$ be a prime. If the spectral-to-tiling direction holds for $\mathbb{Z}_n$ (denoted $S\text{-}T(\mathbb{Z}_n)$), then it holds for $\mathbb{Z}_n \times \mathbb{Z}_p$ (which is isomorphic to $\mathbb{Z}_{np}$).
• Theorem 1.2 (Corollary): For a square-free integer $M = \prod_{i=1}^k p_i$ where the prime factors grow rapidly (specifically $p_{j+1} > \prod_{i=1}^j p_i$), Fuglede's conjecture holds for $\mathbb{Z}_M$.

This result is significant because it provides the first known infinite family of cyclic groups satisfying Fuglede's conjecture that can possess an arbitrary number of distinct prime divisors, provided the primes grow sufficiently fast.

Significance and Claims
The paper claims to resolve the spectral-to-tiling direction for a broad class of cyclic groups that were previously inaccessible to existing methods.

• Inductive Framework: The work develops a robust inductive technique that bridges the gap between small cyclic groups and those with many prime factors, relying on the "large prime" phenomenon.
• Common Spectrum: A key technical insight is Proposition 3.2, which demonstrates that in the presence of a large prime factor, a spectral set must have a common spectrum across all its levels. This structural rigidity is crucial for the inductive step.
• Limitations: The author notes that the square-free hypothesis is essential only in the final step of the proof (Case 2) to ensure a common tiling complement exists for the levels. The method suggests that the result could extend to non-square-free groups if the Coven-Meyerowitz conjecture holds for them and a compatibility argument for complements can be established.
• Context: The paper acknowledges that the tile-to-spectral direction was already known for square-free orders (via Shi and Tijdeman). The novelty lies strictly in the spectral-to-tiling direction.

The author modestly frames the work as an inductive step building upon previous collaborations (Fallon, Kiss, Mayeli) and recent preprints, aiming to provide a foundation for future extensions to non-square-free groups and to address open questions regarding the asymmetry of the conjecture (e.g., groups that are T-S but not S-T).