Nivat's conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $A$ be a finite alphabet and $x\in A^{\mathbb Z^2}$. For $n_1,n_2\in\mathbb N$ write $R=\{0,\ldots,n_1-1\}\times\{0,\ldots,n_2-1\}$ and \[N_x(n_1,n_2)=\left|\{(x_{u+v})_{v\in R}:u\in\mathbb Z^2\}\right|.\] A configuration is periodic if there exists one nonzero vector $p\in\mathbb Z^2$ with $x_{u+p}=x_u$ for every $u$. Two linearly independent period vectors are not required.

Statement recorded in the supplied source.

If $N_x(n_1,n_2)\le n_1n_2$ for at least one pair of positive integers $(n_1,n_2)$, must $x$ be periodic? [S2]

Short English statement

Does having no more rectangular patterns than positions in one rectangle force a nonzero lattice period?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 4600054 (AMR-045-0054). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 C. F. Colle, On Periodic Decompositions, One-sided Nonexpansive Directions and the Nivat Conjecture (2022), Section 1.1, periodicity and pattern complexity; Conjecture 1.3, pp.1--2. https://arxiv.org/pdf/1909.08195.