Year: 2026
Type: preprint
Abstract: We study quantitative edit-distance asymptotics for symbolic codings of irrational rotations $x \mapsto x+α$ on $\mathbb{T}$ in terms of the irrationality exponent $μ(α)$. For the binary coding determined by an interval $[0,β)$, let $d_N(x,y)$ be the edit distance between the codings of the orbits of the points $x$ and $y$ up to time $N-1$. For every $α\notin \mathbb{Q}$, almost every $β\in (0,1)$, and almost every $x,y \in \mathbb{T}$, we determine the edit-distance exponents $\displaystyle (*) \; \; \limsup_{N\to\infty}\frac{\log d_N(x,y)}{\log N} = \frac{ρ(α)}{ρ(α)+1} \; \;$ and $\; \; \displaystyle\liminf_{N\to\infty}\frac{\log d_N(x,y)}{\log N} = \min\Bigl\{\frac{1}{2},\frac{1}{ρ(α)}\Bigr\}$, where $ ρ(α):=μ(α) - 1$. We find a surprising transition at the golden mean $φ= \frac{1+\sqrt{5}}{2}$: If $ ρ(α) \le φ$, then both asymptotics in $(*)$ hold for all $β$. However, for every finite $ρ>φ$, there is an uncountable set of $α$ with $ρ(α)=ρ$ for which both edit-distance exponents are strictly smaller than the corresponding values in $(*)$ for uncountably many $β$. We also derive consequences for aperiodic circle homeomorphisms and Sturmian sequences. For rotations of $\mathbb{T}^d$ coded by boxes, we prove that for almost every rotation vector, the common edit-distance exponent is $d/(d+1)$.
Source: arXiv (Cornell University)
Authors: Best, Andrew, Peres, Yuval
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7202342562
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