arXiv subjects

Thomas Fallon

Publications and source records attributed to Thomas Fallon.

3 recordsLinked to original sources

arXiv2023

Spectral Tile Direction in the Group $\mathbb{Z}_{p^2} \times \mathbb{Z}_{q^2} \times \mathbb{Z}_r$

Let $p$, $q$, and $r$ be distinct primes such that $p^2q^2<r$. We prove that every spectral set in the cyclic group $\mathbb{Z}_{p^2q^2r}$ is a tile. Since the reverse direction is already known, this shows that $\mathbb{Z}_{p^2q^2r}$ is a Fuglede group under this condition. The proof is based on divisibility properties of mask polynomials and on the structure of spectral sets in finite cyclic groups.

math.CA↗

arXiv2021

Spectral sets and tiles in $\mathbb{Z}_p^2 \times \mathbb{Z}_q^2$

In this paper we study Fuglede's conjecture on the direct product of two cyclic groups. After collecting the known results we prove that Fuglede's conjecture holds on $\mathbb{Z}_{pq} \times \mathbb{Z}_{pq} \cong \mathbb{Z}_p^2 \times \mathbb{Z}_q^2$.

math.CA↗

arXiv2019

The Fuglede Conjecture holds in $\F_p^3$ for p=5,7

For p=5,7, we show that a subset $E \subset \F_p^3$ is spectral if and only if E tiles $\F_p^3$ by translation. Additionally, we give an alternate proof that the conjecture holds for p=3.

math.CA↗

Thomas Fallon

Spectral Tile Direction in the Group $\mathbb{Z}_{p^2} \times \mathbb{Z}_{q^2} \times \mathbb{Z}_r$

Spectral sets and tiles in $\mathbb{Z}_p^2 \times \mathbb{Z}_q^2$

The Fuglede Conjecture holds in \(\F_p^3\) for p=5,7