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

TL;DR

This paper proves that in three-dimensional finite fields with p=5 or 7, spectral sets are exactly those that tile by translation, confirming the Fuglede conjecture for these cases, and provides an alternative proof for p=3.

Contribution

The paper establishes the Fuglede conjecture in \\F_p^3 for p=5 and 7, and offers a new proof for p=3, advancing understanding of spectral sets and tiling in finite fields.

Findings

Spectral sets in \\F_p^3 for p=5,7 are exactly the translational tiles.

Confirmed the Fuglede conjecture for p=5,7 in three-dimensional finite fields.

Provided an alternative proof of the conjecture for p=3.

Abstract

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.