A linear programming approach to Fuglede's conjecture in $\mathbb{Z}_p^3$

TL;DR

This paper applies linear programming bounds to Fuglede's conjecture in the finite group _p^3, establishing that certain subset sizes cannot be spectral, thus providing partial evidence towards the conjecture.

Contribution

It introduces a linear programming approach to Fuglede's conjecture in _p^3 and proves a new non-spectrality result for subsets within a specific size range.

Findings

Subsets with size between p^2 - p√p + √p and p^2 are not spectral.

The linear programming method yields partial results supporting Fuglede's conjecture in _p^3.

Abstract

We present an approach to Fuglede's conjecture in $\mathbb{Z}_p^3$ using linear programming bounds, obtaining the following partial result: if $A\subseteq\mathbb{Z}_p^3$ with $p^2-p\sqrt{p}+\sqrt{p}<|A|<p^2$, then $A$ is not spectral.