L. Sadun — Problem

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.

Matching rules specify permitted local arrangements of finitely many labeled prototile types. For a type $t$, let $d(t)$ be its number density per unit volume in the resulting tilings. If $R$ is a finite union of tiles, write $N_t(R)$ for the number of its type-$t$ tiles and \[D_t(R)=|N_t(R)-d(t)\operatorname{vol}(R)|.\] Write $|\partial R|$ for $(d-1)$-dimensional boundary measure. Condition (A), low fluctuation, requires a constant $c_t$ for each tile type, uniform over finite regions $R$ in the tilings, such that $D_t(R)\leq c_t|\partial R|$. Condition (B) requires the same bound for every finite patch satisfying the local matching rules, including patches that need not extend to a complete admissible tiling. The densities and constants belong to the chosen rule system, rather than varying with $R$.

Statement recorded in the supplied source.

Find matching rules in dimension $2$ satisfying condition (A), without requiring condition (B). [S2]

Short English statement

Find planar matching rules giving a boundary-size discrepancy bound for finite regions of complete tilings.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 6600018 (AMR-065-0018). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Faustin Adiceam (compiler), Open Problems and Conjectures related to the Theory of Mathematical Quasicrystals, arXiv:1604.06280v2 (2016), Section 2.7.2, condition (A) and Problem 2.7.4, p. 10. https://arxiv.org/pdf/1604.06280.