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

Romanos Diogenes Malikiosis

Session: Frames, Riesz bases, and related topics (Jorge Antezana)

Keywords: Fuglede's conjecture, spectral sets, tiling, blocking sets, finite fields, projective plane

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

Abstract:

Delsarte's method on linear programming bounds is a very powerful tool which provides an upper bound on the size of a set $A$ in an additive group $G$, whose difference set $A-A$ avoids a given set $E$. This tool may have limitations, but has been used successfully in various settings, most notably in the sphere packing problem in 8 and 24 dimensions.

Here, we will present an application of this method to Fuglede's conjecture in $G=\mathbb{Z}_p^3$, providing the following partial result: a set $A\subset G$ with cardinality
$$p(p-\sqrt{p}-\frac{1}{\sqrt{p}})<|A|<p^2,$$
cannot be spectral.

Submission Number: 47