arXiv:2608.11763v1

Edit distance exponents for irrational rotations

Andrew Best, Yuval Peres

math.DSmath.NT37A4437A3511K50

Paper metadata

Original submission: August 12, 2026
Claimed MSC: 37A44, 37A35, 11K50
arXiv categories: math.DS, math.NT
Paper license: arXiv non-exclusive distribution license ↗
Comments: 47 pages, 2 figures

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 $μ(α)$ , the supremum of $μ\in \mathbb{R}$ for which the inequality $0 < |α- p/q| < q^{-μ}$ has infinitely many solutions. For the binary coding determined by an interval $[0,β)$ , let $\mathcal{W}_N$ be the set of length- $N$ words arising from all initial points $x$ under $x \mapsto x+α$ . We develop new techniques for estimating edit distance and compute the growth exponents of the edit-distance diameter $\mathrm{diam}_E(\mathcal{W}_N)$ . For every $α\notin \mathbb{Q}$ and almost every $β\in (0,1)$ , we show that $\displaystyle (*) \quad \limsup_{N\to\infty}\frac{\log \mathrm{diam}_E(\mathcal{W}_N)}{\log N} = \frac{μ(α)-1}{μ(α)},$ and the corresponding $\liminf$ equals $1/2$ . When $μ(α)-1$ is at most the golden mean $\varphi$ , the asymptotics $(*)$ hold for all $β$ . However, for $μ>1+\varphi$ , there is an uncountable set of $α$ with $μ(α)=μ$ for which the edit-distance exponents are strictly smaller than $(*)$ 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 $d/(d+1)$ . Finally, we raise the question of estimating edit-distance exponents for more general dynamical systems.

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Audited against arXiv v1

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

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Detailed mathematical audit

Generated August 18, 2026

01Statements4 reported findingsCorrect

The four one-dimensional edit-distance exponents, their all- $\beta$ golden-mean regime, the sharp exceptional-parameter bounds, the circle-map and Sturmian consequences, and the almost-everywhere $d$ -torus exponent are correct. The continued-fraction, matching, probabilistic, and discrepancy estimates give the stated exponents with the required quantifiers. One proof of an auxiliary upper bound omits the regime in which a convergent denominator exceeds the averaging length, but the omitted case has a verified stronger estimate and does not change Proposition 5 or any other statement.

Theorem 1 and Corollary 2Correct

Typical coding parameters and the multifractal dimension formula

Pages 2 and 8–23 · Theorem 1, Corollary 2, and Sections 3–7 · arXiv:2608.11763v1

The approach-and-follow estimate gives the universal upper bounds $\overline\Gamma(\alpha,\beta)\leq\frac{\mu(\alpha)-1}{\mu(\alpha)},\qquad \underline\Gamma(\alpha,\beta)\leq\frac12.$ The common-subsequence lemma converts small edit distance into concordance of most short blocks. Away from the summable exceptional sets for $\operatorname{dist}(\beta,\Lambda(q_m))$ and $\lVert q_m\beta\rVert$ , Lemmas 6.2 and 7.4 give the matching diameter and typical-pair lower bounds. Weyl equidistribution of $(q_m\beta)$ on a subsequence realizing $\mu(\alpha)$ supplies the typical-pair limsup lower bound. The positive-measure subsequential conclusions become almost-everywhere conclusions because the exponent functions are invariant under the ergodic independent-shift $\mathbb Z^2$ -action. Finally, $\gamma=(\mu-1)/\mu$ is equivalent to $\mu=1/(1-\gamma)$ , so Güting's exact-exponent dimension $2/\mu$ gives $2-2\gamma$ , including the Liouville endpoint by its known zero dimension.

Full paper, version 1 ↗

Theorems 3–4 and Proposition 5Correct

Uniform lower bounds, the golden-mean threshold, and exceptional parameters

Pages 3 and 24–36 · Theorems 3–4, Proposition 5, and Sections 8–9 · arXiv:2608.11763v1

The signed clean-block argument in Lemma 8.2 and its probabilistic forms control every coding length $\beta$ , with no metric exclusion. Optimizing the convergent scale gives the threshold $\rho^2-\rho-1=0$ , hence $\rho=\varphi$ . For $\rho>\varphi$ , the round-and-synchronize estimate applies whenever $\beta$ is eventually $O(q_{m+1}^{-1})$ -close to the length- $q_m$ orbit. The nested construction in Proposition 9.3 provides a perfect set of such parameters, while prescribed partial quotients give uncountably many $\alpha$ with steady growth $q_{m+1}=q_m^{\rho+o(1)}$ . The optimizations then yield exactly $\frac{\rho^2}{2\rho^2-1}\quad\text{and}\quad\frac{\rho}{\rho^2+\rho-1}.$ Proposition 5 follows from the same synchronization lemma; the missing case in its printed proof is repaired in Part 2.

Full paper, version 1 ↗

Corollary 6Correct

Transfer to aperiodic circle homeomorphisms

Pages 3 and 36–37 · Corollary 6 and its proof · arXiv:2608.11763v1

For a dense-orbit aperiodic circle homeomorphism, choose the orientation-preserving conjugacy $h$ to the rotation $R_{\omega(f)}$ with $h(0)=0$ . Then $\mathbf 1_{[0,\eta)}(f^n x)=\mathbf 1_{[0,h(\eta))}(R_{\omega(f)}^n h(x)),$ so the word sets agree exactly and Theorem 1 applies with $\beta=h(\eta)$ . The unique invariant measure satisfies $\nu(h^{-1}E)=\mathcal L(E)$ , giving the stated $\nu$ -almost-everywhere parameter result. If $r>\mu(\omega(f))$ , choose a Diophantine exponent strictly between $\mu(\omega(f))$ and $r$ ; the cited smooth-conjugacy theorem gives a $C^1$ conjugacy, which transfers null sets to Lebesgue-null sets as claimed.

Katznelson–Ornstein, smoothness of the circle conjugacy ↗

Theorem 7Correct

The common exponent for typical rotations of $\mathbb T^d$

Pages 3–4 and 38–42 · Theorem 7 and Theorems 11.7, 11.10 · arXiv:2608.11763v1

For a rotation vector with discrepancy excess $\Delta_q=q^{o(1)}$ , the covering consequence of discrepancy finds an orbit shift within distance $O((\Delta_q/q)^{1/d})$ . Counting visits to the $O(\epsilon^{1-d})$ boundary cubes gives the upper exponent $d/(d+1)$ . Conversely, equal length- $k$ box codings force the two points within $O(\Delta_k/k)$ ; Lemma 4.3 and a union bound therefore give a summable small-edit probability at the optimized scale $k=N^{1/(d+1)}$ . Borel–Cantelli yields the typical-pair lower exponent $d/(d+1)$ , and the ordering between typical and diameter exponents forces all four exponents to coincide. Beck's metric discrepancy theorem supplies $\Delta_q=q^{o(1)}$ for almost every rotation vector.

Beck, metric discrepancy of Kronecker sequences ↗

02Proofs4 reported findingsContains incorrect or incomplete proofs

The central proof chains for the one-dimensional typical results, the golden-mean transition, the sharp exceptional constructions, circle conjugacy, and multidimensional discrepancy bounds are correct and complete. The proof of Proposition 9.9, used for Proposition 5(a), incorrectly asserts that an adapted exponent $\theta$ is at most one and therefore omits the possible large-denominator regime. That regime is repaired directly from Lemma 9.1 and in fact gives a stronger estimate.

Core one-dimensional proof chainCorrect and complete

Approach-and-follow, concordance, and clean-block estimates close

Sections 3–8 · arXiv:2608.11763v1

Lemma 3.2 balances the time needed to approach a second orbit against visits to the two coding boundaries. Lemma 4.3 correctly extracts at least $N-3k\ell$ concordant starting positions from a common subsequence of defect at most $\ell$ . Lemma 6.2 uses the separation of $\beta$ from $\Lambda(q_m)$ to produce a discordant set of measure greater than $3N^{-\epsilon}$ , and Lemma 7.4 obtains the complementary typical-pair estimate by comparing the number of ones in $q_m$ -blocks. In Section 8, opposite signs of clean blocks force matching-shift variation, while missing a sign forces many non-clean blocks; Lemma 7.2 controls both by the edit defect. All exponent inequalities, summability conditions, endpoint conventions for $\mu(\alpha)=\infty$ , and uses of the almost-everywhere invariant exponents are consistent.

Round-and-synchronize lemmaCorrect and complete

The periodic rounding and drift corrections give the stated edit bound

Pages 31–32 · Lemma 9.1 · arXiv:2608.11763v1

Rounding $\beta$ to $j_0/q_m$ changes at most $4b$ symbols in each block of length $q_{m+1}$ . Rounding $\alpha$ to $p_m/q_m$ produces a $q_m$ -periodic word. Each accumulated displacement of $1/q_m$ is synchronized by deleting $q_{m-1}$ symbols, using $p_{m-1}q_m-p_mq_{m-1}=(-1)^m$ , and consecutive corrections are at least $q_{m+1}$ positions apart. The common-subsequence construction therefore proves $\operatorname{diam}_E(\mathcal W_N)\leq q_m+C(q_{m-1}+b)\left\lceil\frac{N}{q_{m+1}}\right\rceil$ for both signs of $\alpha-p_m/q_m$ .

Proof of Proposition 9.9Incorrect as written · verified repair

The adapted convergent can be larger than the averaging length

Page 36 · proof of Proposition 9.9, immediately after choosing $q_m<N^{1/3}\leq q_{m+1}$ · arXiv:2608.11763v1

The proof writes $q_{m+1}=N^\theta$ and asserts $\theta\in[1/3,1]$ . There is no upper bound $q_{m+1}\leq N$ ; for a large partial quotient one can have $q_m<N^{1/3}$ but $q_{m+1}>N$ , so $\theta>1$ . The subsequent estimate $q_mN^{1-\theta}$ also cannot absorb the leading $q_m$ term in that regime. The repair is verified and stronger: if $q_{m+1}>N$ , Lemma 9.1 with $b=2$ gives $\operatorname{diam}_E(\mathcal W_N)\leq q_m+C(q_{m-1}+2)\left\lceil\frac{N}{q_{m+1}}\right\rceil=q_m+C(q_{m-1}+2)\ll q_m<N^{1/3}.$ If $q_{m+1}\leq N$ , then $\theta\in[1/3,1]$ and the paper's two-case optimization applies unchanged, giving $O(N^{2/3})$ . Thus Proposition 9.9 and Proposition 5(a) are correct, but the printed proof omits a necessary case.

Multidimensional proof chainCorrect and complete

Discrepancy and transference inputs are applied with matching scales

Sections 11.2–11.4 · arXiv:2608.11763v1

The radius in Lemma 11.3 is chosen so that a slightly enlarged cube has $q$ times its volume greater than $\Delta_q$ , and passage to the closed cube is valid. Lemma 11.5 counts disagreements in the boundary neighborhood with the correct factor $\epsilon^{1-d}$ . For the lower bound, the box in Lemma 11.8 has volume $\Delta_k/(k\beta_i)$ , so discrepancy forces a coding mismatch whenever the points are farther than $C_3\Delta_k/k$ . The optimized Borel–Cantelli calculation then yields $d(1-\delta)/(1+d(1-\delta))$ . In the non-singular extension, the cited homogeneous-to-inhomogeneous transference gives the required covering radius, while open cubes of radius $\psi_\alpha(q)/2$ contain at most one point of a length- $q$ orbit segment; the resulting subsequential liminf and typical limsup bounds follow.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2608.11763v1
Authors listed: Andrew Best, Yuval Peres
Audit date: August 18, 2026

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Edit distance exponents for irrational rotations

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Typical coding parameters and the multifractal dimension formula

Uniform lower bounds, the golden-mean threshold, and exceptional parameters

Transfer to aperiodic circle homeomorphisms

The common exponent for typical rotations of Td\mathbb T^dTd

Approach-and-follow, concordance, and clean-block estimates close

The periodic rounding and drift corrections give the stated edit bound

The adapted convergent can be larger than the averaging length

Discrepancy and transference inputs are applied with matching scales

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The common exponent for typical rotations of $\mathbb T^d$