Results on tilings, the arithmetic word rule, tiling equations, stacking and the passage to three dimensions, with their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Elementary facts on tilings — Lemma 1.2. Tiles and fully periodic tiling sets are preserved by translations and isomorphisms; A (+) H = G for a subgroup H exactly when A meets every coset once; fully periodic sets are finite unions of cosets of a finite-index subgroup. No hypothesis.
• Newman's theorem — Theorem 2.1. Every tiling set of Z by a finite tile is invariant under a translation by some p >= 1. No hypothesis.
• Subgroups of Z^d — Lemma 3.2. A subgroup of Z^d of index M contains M Z^d; a subgroup containing d linearly independent vectors has finite index. No hypothesis.
• Unit-cube thickening — Proposition 3.3. A tiling set of Z^d by T is an almost-everywhere tiling set of R^d by T + [0,1]^d, fully periodic if invariant under a finite-index subgroup. No hypothesis.
• A tile in Z^3 with no periodic tiling (OpenAI) — Theorem 4.1. A finite T in Z^3 tiles Z^3 with no tiling set invariant under a finite-index subgroup, and T + [0,1]^3 tiles R^3 almost everywhere with no tiling set invariant under a lattice of full rank; proved in the last lesson. No hypothesis.
• Global affine approximation — Lemma 2.1. An array with the line rule (p > 200) agrees with a nonzero affine form An + Bm + C over F_p wherever the form is nonzero. No hypothesis.
• Vertical nondegeneracy — Lemma 3.1. The columns are nonconstant exactly when B != 0, and then every positive vertical period is divisible by p. No hypothesis.
• The vertical-period obstruction — Proposition 4.1. No array with the line rule and no constant column has a positive vertical period. No hypothesis.
• The last-nonzero-digit model — Lemma 5.1. The array W(n,m) = f(m), with f the last nonzero digit in base p, satisfies the line rule and every column takes every value. No hypothesis.
• A cyclic finite group — Lemma 1.2. V = Z/p^4 x prod (Z/r_i^2 x Z/a_i x Z/b_i) is cyclic. No hypothesis.
• Invariance tiles — Lemma 3.1. For a channel i and a shift sigma there is a tile F with A (+) F = G exactly when c_i(b - sigma) = c_i(b) for all bases b, for every graph A. No hypothesis.
• Exclusion of simultaneous activity — Lemma 4.1. For a cycle of channels with chosen labels, finitely many tiles are tiled by a graph exactly when at every x one of the labels is inactive. No hypothesis.
• Activation fibre criterion — Lemma 3.1. A graph tiles with an activation tile exactly when the batch maps are bijections at every target base. No hypothesis.
• Ordinary activation — Lemma 4.1. In every common solution every ordinary channel has an active label at every point. No hypothesis.
• Seed activation — Lemma 4.2. In every common solution each seed is active at some point. No hypothesis.
• No fully periodic common solution — Proposition 5.1. No common solution of the system is invariant under a subgroup of finite index. No hypothesis.
• Block inverse — Lemma 2.1. The map (l, delta) -> (C(k - l) + e_delta, T(x_1 - h_1)(k - l)) is a bijection from K x Delta onto P x K. No hypothesis.
• A system with a common solution and no periodic one (OpenAI) — Theorem 6.2. Finitely many finite tiles in Z^2 x V, V finite cyclic, have a common solution but no fully periodic common solution. No hypothesis.
• Partitions with full difference sets — Lemma 1.1. For every s and every finite set of primes there are a prime q outside it and a partition of Z/q into s sets E with E - E = Z/q. No hypothesis.
• Stacking — Proposition 2.1. Finite tiles F_1, ..., F_s have a common solution exactly when the stacked tile tiles H x Z/q, and full periodicity passes in both directions. No hypothesis.
• One tile with a cyclic finite coordinate — Corollary 3.1. A finite tile of Z^2 x Z/Q has a tiling set but no fully periodic one. No hypothesis.
• Rigid representatives — Lemma 1.1. For some m >= 2 there is a transversal modulo m whose points other than the marked one have every difference d with |d| <= m outside m Z^3. No hypothesis.
• Existence of tilings — Proposition 3.1. The tile T tiles Z^3 and its thickening tiles R^3 almost everywhere. No hypothesis.
• Grid centers — Lemma 4.2. In a periodic almost-everywhere tiling by the two constituent shapes, the centers form a translate of m Z^3. No hypothesis.
• No fully periodic tiling of R^3 — Proposition 5.1. No almost-everywhere tiling of R^3 by the thickening of T has a tiling set invariant under a lattice of full rank. No hypothesis.

Reading order

Seven lessons. The first sets up tilings and periodicity, proves Newman's theorem and states the theorem; the second proves the obstruction for the arithmetic word rule; the third, fourth and fifth build the tiling equations in Z^2 x V, exclude periodic solutions and construct a common solution; the sixth stacks the system into one tile; the seventh passes to Z^3 and R^3 and proves the theorem. Everything used is proved in the course apart from basic algebra, linear algebra and Lebesgue measure from the core courses; the planar periodicity theorem is stated as background and not used.