The Algebraicity Problem for Hard-Core Entropy Constants on the Discrete Hypertori

Abstract: We use tools and techniques from $p-$adic analysis and algebraic number theory to study the algebraicity of the hard square entropy constant and its high dimensional analogues. Specifically, We study arithmetic properties of $a_d(n)$, the number of independent sets in the $d$-dimensional discrete torus, and the associated entropy constants $κ<em>d=\lim</em>{n\to\infty}a_d(n)<sup>{1/n<sup>d}$. It is not known whether $κ_d$ is algebraic or transcendental for $d&gt;1$. Using the fact that the sequence $a_d(p<sup>k)$ converges $p-$adically for every prime $p$, we collection of criteria for the algebraicity of $κ_d$ and bound the number of possible values of prime powers $p<sup>k$ for which $a_d(p<sup>k)=κ_d<sup>{p<sup>{kd}}$.