Moduli of toric tilings into bounded remainder sets and balanced words

Abstract:

The moduli space $\mathcal {M}_{\mathrm {til}}$ is constructed for the family $\mathbb {T}_{\mathrm {til}}$ of parallelotope tilings \[ \mathbb {T}^{D}_{c,\lambda }=\mathbb {T}^{D}_0 \sqcup \mathbb {T}^{D}_1 \sqcup \dots \sqcup \mathbb {T}^{D}_D \] of the torus $\mathbb {T}^D=\mathbb {R}^D/\mathbb {Z}^D$ of arbitrary dimension $D$ into bounded remainder sets $\mathbb {T}^{D}_k$. By using these tilings, the Hecke theorem on the distribution of fractional parts on the circle is extended to the tori $\mathbb {T}^D$: the deviation of the distribution of points of an orbit with respect to the translation $S_{\beta } : x \rightarrow x+\beta \bmod \mathbb {Z}^D$ of the torus $\mathbb {T}^D$ by an arbitrary vector $\beta =\frac {1}{n}(\lambda c+l)$ is estimated in terms of the moduli $(c,\lambda )\in \mathcal {M}_{\mathrm {til}}$, where $l$ lies in the cubic lattice $\mathbb {Z}^D$.

The color and frequency universality is proved for the toric tilings $\mathbb {T}^{D}_{c,\lambda }$ from the family $\mathbb {T}_{\mathrm {til}}$ and it is shown how these tilings can be used to generate $\kappa$-balanced words $w$ in the alphabet $\mathcal {A}=\{0,1, \dots ,D \}$ with $\kappa =2$ for $D=2$ and $\kappa =3$ for $D\geq 3$.