Local fluctuations, majorization, and nontransitivity in cube tilings of ${\mathbb{Z}}^d$

Abstract

Let $\mathcal{P}$ be a partition of ${\mathbb{Z}}^d$ into translates of the discrete unit cube $\{0,1{\}}^d$. For a uniformly chosen $\epsilon \in \{0,1{\}}^d$, we count the cubes of $\mathcal{P}$ entirely contained in the observation window $[0,3{]}^d+\epsilon$. The resulting score has mean $(3/2{)}^d$ for every partition, while its distribution reflects the geometry of the tiling. Comparing two such scores, with the shifts chosen independently, defines a game in which the larger score wins. In dimension $d=4$, we construct a geometric realization of the Steinhaus--Trybuła paradox: three periodic partitions form a cycle in which each beats the next with probability strictly greater than $1/2$. We describe local configurations by finite cube tiling codes and prove that every resulting list of counts is majorized by that of the simple code $C_d=\{0,2{\}}^d$. Nevertheless, every non-simple code differing from $C_d$ in only two coordinate directions beats it, with a sharp lower bound on the winning margin. We also show that the ratio of wins to losses of $C_d$ has limit superior at most $1$ as the dimension tends to infinity, uniformly over opposing codes. For $4{\mathbb{Z}}^d$-periodic tilings, changing the window size can repeatedly reverse the winner, but the winning margin eventually stabilizes separately for side lengths $4m+1$ and $4m+3$.

arXiv comment30 pages, 4 figures