Obtaining the Chamanara Surface from the van der Corput sequence

TL;DR

The paper addresses how 4-regular sequence graphs derived from deterministic sequences can be embedded into translation surfaces. It proves an exact torus embedding for Kronecker graphs in the case $N = \pi(1) + \pi(N-1)$ and shows a robust torus embedding up to a single edge for other $N$, while demonstrating a Chamanara-surface embedding for binary van der Corput graphs when $N = 4^m$ (with a one-edge deletion otherwise). The authors connect these embeddings to interval exchange transformations, offering a unifying perspective and outlining a general theory relating sequence structure to surface embeddings via IET dynamics. These results illuminate the geometric structure underlying deterministic sequences and suggest broader applicability to surface topology and translation surfaces, including finite and infinite-type cases via $\alpha = 1/b$ generational families.

Abstract

We investigate a family of $4$-regular graphs constructed to test for the presence of combinatorial structure in a sequence of distinct real numbers. We show that the graphs constructed from the Kronecker sequence can be embedded into the torus, while the graphs constructed from the binary van der Corput sequence can be embedded into the Chamanara surface, in both cases with the possible removal of one edge. These results allude to a general theory of sequence graphs which can be embedded into particular translation surfaces coming from interval exchange transformations.

Figures (6)

• Figure 1: Left: Kronecker sequence graph with $N=8$; colors show the two Hamiltonian cycles. Right: Kronecker sequence graph with $N = 971$ resembles a torus besides one rogue edge.
• Figure 2: Left: van der Corput sequence graph at $N = 8$. Right: van der Corput sequence graph at $N = 1024$ resembles a surface.
• Figure 3: The Chamanara Surface
• Figure 4: Plots of $(a_i, a_{i+1})$ as $i$ varies for Kronecker sequence on the left and van der Corput on the right.
• Figure 5: The sequence generated by iterating the transform $T_{\pi, \lambda}$ on the left produces surface-like sequence graphs (with $N=1000$) on the right.

Theorems & Definitions (12)

• Example 1.1
• Theorem 1.2: label=thm:kron-nice, restate=thmkronnice
• Theorem 1.3: label=thm:kron-hard, restate=thmkronhard
• Example 1.4
• Theorem 1.5: label=thm:vdc-nice, restate=thmvdcnice
• Theorem 1.6: label=thm:vdc-hard, restate=thmvdchard
• proof : Proof of \ref{['thm:kron-nice']}
• Lemma 2.1: label=lem:vdc-succ
• proof
• proof : Proof of \ref{['thm:vdc-nice']}