A Modular Structure Theorem for Minimal Periodic Decompositions and Periodicity of Configurations with $P_{\eta}(4,n) \leq 4n$

Abstract

Nivat's conjecture asserts that every two-dimensional configuration $\eta : \mathbb{Z}^2 \to \mathcal{A}$ whose rectangular pattern complexity satisfies $P_{\eta}(k,n) \leq kn$ for some $k,n \in \mathbb{N}$ is periodic. A theorem of Cyr and Kra \cite{CyrKra16} establishes the conjecture in the short-rectangle case $P_{\eta}(k,n) \leq kn$, with $k \leq 3$. Using the algebraic framework of Kari-Szabados \cite{KariSzabados20} and recent advances on periodic decompositions and one-sided nonexpansive directions \cite{Colle23,Colle22}, we extend the Cyr-Kra result to the case $P_{\eta}(4,n) \leq 4n$: every configuration satisfying this complexity bound is periodic. The key new ingredient is an intermediate structural theorem of independent interest: for any non-periodic configuration with low convex pattern complexity and integer-valued alphabet $\mathcal{A}$ contained in $\mathbb{Z}_+$, there exist a configuration $\vartheta$ in the orbit closure of $\eta$, a $\mathbb{Z}$-minimal periodic decomposition $\vartheta = \vartheta_1+\cdots+\vartheta_m$, a prime $p \in \mathbb{N}$ with $\mathcal{A} \subset [[p]]$, and pairs of disjoint half-planes $U_i,V_i \subset \mathbb{Z}^2$ such that the reductions modulo $p$ of the components $\vartheta_i$ are fully periodic on $U_i$ and on $V_i$ simultaneously, for each $1 \leq i \leq m$.