Edit distance exponents for irrational rotations

Abstract

We study quantitative edit-distance asymptotics for symbolic codings of irrational rotations x ↦ x+α on 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∉ ℚ, almost every β∈ (0,1), and almost every x,y ∈ 𝕋, we determine the edit-distance exponents (*) limsup_N→∞(log d_N(x,y))/(log N) = ρ(α)/(ρ(α)+1) and liminf_N→∞(log d_N(x,y))/(log N) = min{1/2,1/ρ(α)}, where $ ρ(α):=μ(α) - 1$. We find a surprising transition at the golden mean φ= (1+√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 𝕋^d coded by boxes, we prove that for almost every rotation vector, the common edit-distance exponent is d/(d+1).

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BibTeX RIS

BibTeX

@misc{best2026edit,
  title = {{Edit distance exponents for irrational rotations}},
  author = {Andrew Best and Yuval Peres},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.11763},
  eprint = {2608.11763},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.11763},
}

RIS

TY  - UNPB
TI  - Edit distance exponents for irrational rotations
AU  - Andrew Best
AU  - Yuval Peres
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2608.11763
UR  - https://arxiv.org/abs/2608.11763
AB  - We study quantitative edit-distance asymptotics for symbolic codings of irrational rotations x ↦ x+α on 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∉ ℚ, almost every β∈ (0,1), and almost every x,y ∈ 𝕋, we determine the edit-distance exponents (*) limsup_N→∞(log d_N(x,y))/(log N) = ρ(α)/(ρ(α)+1) and liminf_N→∞(log d_N(x,y))/(log N) = min{1/2,1/ρ(α)}, where $ ρ(α):=μ(α) - 1$. We find a surprising transition at the golden mean φ= (1+√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 𝕋^d coded by boxes, we prove that for almost every rotation vector, the common edit-distance exponent is d/(d+1).
ER  -

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