arXiv:2605.30606v2

Transcendence and measures via the refined Diophantine exponent

Quang-Khai Nguyen

math.NT11J6811J8211J8737B10

Paper metadata

Original submission: May 28, 2026
Latest revision: August 1, 2026 · version 2
Claimed MSC: 11J68, 11J82, 11J87, 37B10
arXiv categories: math.NT
Paper license: CC BY 4.0 ↗
Comments: Typos fixed

Abstract

In 2007, Adamczewski and Bugeaud introduced the notion of the Diophantine exponent of an infinite word as a quantitative measure of repetition, leading to new transcendence results for real numbers whose expansions in an integer base are sufficiently simple. In the present article, we introduce the refined Diophantine exponent, which detects weaker forms of repetition while preserving the full strength of the classical approach. This new exponent applies in situations where repetition is partially obscured by some form of noise. Related ideas already appear in the work of Corvaja and Zannier in 2002 and, more recently, in the works of Kebis, Luca, Ouaknine, Scoones, and Worrell. Our approach provides a unified framework that recovers and extends these results, as well as those of Adamczewski and Bugeaud. We also develop quantitative refinements of this method, leading to results about transcendence measures. The recent breakthrough of Bell, Diller, and Jonsson in the context of algebraic dynamics is partly based on a similar idea, which also served as a motivation for the present work.

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Audit summary

Audited against arXiv v2

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.

Current report

Detailed mathematical audit

Generated August 15, 2026

01Statements4 reported findingsContains unsupported statements

The transcendence criterion and the quantitative results in Theorems A–C are correct after the harmless notation repairs recorded below. Theorem D and the stronger auxiliary criterion Theorem 6.5 are not verified: their proof uses a lower-bound lemma whose printed hypotheses are strictly broader than those of the cited source and under which the lemma is false. No counterexample to Theorem D itself was found.

Theorem ACorrect

Refined-Diophantine-exponent transcendence criterion

Pages 4 and 18–23 · Theorem A and Section 3.2 · arXiv:2605.30606v2

The sparse-mismatch data give the required Subspace-Theorem inequality. The stabilized linear relation is then separated into its widely spaced polynomial blocks by Lenstra's lacunary-factor lemma, and the limiting relation forces the sum either to lie in $K$ or to be transcendental. The two evaluation-point and linear-form subscripts corrected below are uniquely determined notation errors and do not change the argument.

Theorem B and Corollary B.1Correct

Quantitative transcendence measure and Mahler classification

Pages 5–6 and 23–34 · Theorem B, Corollary B.1, and Section 4 · arXiv:2605.30606v2

The quantitative Subspace Theorem, Siegel-lemma construction, Liouville lower bounds, and height estimates combine with the bounded-density repetition hypothesis to give the displayed exponent $(2d)^{c(\log 4d)(\log\log 4d)}.$ The count of exceptional subspaces and the number of relevant points in each subspace have the required logarithmic sizes. The Mahler-classification trichotomy follows from these bounds and the standard height interpretation of $\omega_d$ .

Theorem CCorrect

Lacunary-series dichotomy and quantitative bounds

Pages 6 and 34–38 · Theorem C and Section 5 · arXiv:2605.30606v2

When the quotient of successive support indices is unbounded, truncations give arbitrarily strong approximations of bounded degree. In the bounded-quotient case, the modified quantitative argument and two applications of the lacunary-factor lemma exclude every $U_d$ class and yield $\omega_d(\xi)\leq(2d)^{c\log\log(4d)}$ . Reading the proof with the full coefficient sequence, as specified in the typo correction below, makes the displayed decompositions consistent.

Theorem 6.5 and Theorem DNot able to verify

Exclusion of $U_1$ and $U_2$ for periodic-prefix approximants

Pages 7 and 38–41 · Theorems 6.5 and D · arXiv:2605.30606v2

The intended height-growth argument depends on Lemma 6.4 to give a positive lower bound for the distance from the coefficient series to each eventually periodic approximation. Lemma 6.4 is false for the arbitrary digit words permitted in its printed statement; the cited Adamczewski–Cassaigne result instead assumes the greedy $\beta$ -expansion. The $k$ -bonacci and Sturmian papers cited here verify transcendence, but they do not verify the stronger $S$ - or $T$ -number conclusion. No independent proof of that conclusion under the stated hypotheses, and no counterexample to it, was established in this audit.

Adamczewski–Cassaigne, Lemma 8.2 ↗

02Proofs9 reported findingsContains incorrect or incomplete proofs

The proofs of Theorems A–C are complete after local notation and rounding corrections. The proof of Theorem 6.5, and hence the stated proof of Theorem D's $S$ / $T$ classification, relies on a false generalization of a cited separation lemma. Its height argument also assumes consecutive approximants are distinct without proof. In the Sturmian case, ordinary equidistribution does not supply the uniform shrinking-interval estimate asserted in the text, although bounded type gives a standard repair to that latter step.

Lemma 6.4Incorrect as written

The cited separation bound is false for arbitrary digit words

Page 38 · Lemma 6.4 · arXiv:2605.30606v2

The paper removes the greedy-expansion hypothesis from Adamczewski–Cassaigne's Lemma 8.2. Under the enlarged hypotheses, cancellation can make the two represented numbers equal. Take $\beta=2$ , $U=1$ , $V=01$ , so $b=UV^\infty=101010\ldots$ , and take $j=3$ . Let $a_i=b_i$ except that $a_2=0$ and $a_3=2$ . Then $a_0=b_0$ , $a_1=b_1$ , and $a_2\neq b_2$ , exactly as required, but $\sum_{i\geq0}(a_i-b_i)2^{-i}=-2^{-2}+2\cdot2^{-3}=0.$ Thus $\xi=\alpha$ , contradicting the claimed strictly positive lower bound. The cited source assumes that the digits of $\xi$ form its greedy $\beta$ -expansion and uses the corresponding greedy-tail inequality. No replacement separation theorem for the arbitrary words used in Theorem 6.5 is supplied.

Adamczewski–Cassaigne, original Lemma 8.2 ↗

Claim 6.6Incomplete as written

The height lower bound assumes two approximants are distinct

Pages 38–39 · proof of Claim 6.6 · arXiv:2605.30606v2

The proof applies Liouville's inequality to $\alpha_n$ and $\alpha_{n-1}$ without showing that $\alpha_n\neq\alpha_{n-1}$ . Equality of two different eventually periodic digit words is possible in a non-integer base, and is also possible with the redundant digit alphabet allowed when $\beta$ is an integer. Convergence to the assumed transcendental $\xi$ shows only that infinitely many distinct values occur, not that consecutive ones differ. With a valid lower separation bound one could select first distinct successors and control the skipped range, but that repair presently depends on replacing the false Lemma 6.4.

Full paper, version 2 ↗

Theorem D · Sturmian caseIncomplete as written · verified repair of this step

Ordinary equidistribution is not uniform for the shrinking intervals used

Pages 40–41 · proof of Theorem D for Sturmian words · arXiv:2605.30606v2

For each $n$ , the mismatch set is the union of two intervals of length $\|u_n\theta\|$ , and the proof asserts from equidistribution alone that its number of visits up to the varying endpoint $v_n$ is $\asymp v_n\|u_n\theta\|$ . Equidistribution gives an asymptotic for each fixed interval; it does not by itself give a uniform estimate for intervals shrinking with $n$ and endpoints depending on $n$ . This particular step can be repaired using the bounded-partial-quotient hypothesis: rotations of bounded type have return gaps $O(1/|I|)$ for every interval $I$ , by the continued-fraction net/three-gap estimate. Since $v_n$ is maximal before the fixed mismatch threshold is exceeded, this gives $v_n=O(1/\|u_n\theta\|)=O(u_n)$ , which is the only conclusion subsequently used. This repair does not resolve the separate Lemma 6.4 defect.

Lemma 2.20Minor formal correction

The chosen endpoint and mismatch bound require rounding and slack

Pages 15–16 · final paragraph of Lemma 2.20 · arXiv:2605.30606v2

The proof sets $t_n=\lfloor\rho q_n\rfloor-1$ and $\delta=\rho/\nu$ . The latter need not be an integer, while the former need not satisfy $t_n\geq\rho s_n$ when $s_n=q_n-1$ ; for $1<\rho<2$ this fails whenever $\{\rho q_n\}>\rho-1$ . Fix the target $\rho$ , run the counting estimate with any $\rho_0>\rho$ , and set $t_n=\lfloor\rho_0q_n\rfloor-1$ and an integer $\delta>\rho_0/\nu$ . For all sufficiently large $n$ , $t_n\geq\rho s_n$ , and the mismatch count is bounded by $\delta$ . This is a local parameter repair and leaves Proposition 2.17 unchanged.

Theorem A and Claim 4.16Typo

The special linear form and auxiliary polynomials are evaluated at the wrong symbol

Pages 20, 22, and 30 · proof of Theorem A and Claim 4.16 · arXiv:2605.30606v2

On page 20 the exceptional form must be $L_{3,v}$ , not $L_{3,w}$ , because the next bullet makes every $(i,w)\neq(3,v)$ a coordinate form and Equation (3.4) uses the exceptional form only at $v$ . On pages 22 and 30, the displayed polynomials have positive powers and satisfy $P_n(\beta^{-1})=\beta^{-s_n}\,(\text{displayed linear relation})=0,$ not $P_n(\beta)=0$ . Replace every evaluation at $\beta$ in these lacunary-polynomial steps by $\beta^{-1}$ . Height and root-of-unity properties are unchanged, so the degree-gap arguments remain valid.

Theorem CTypo

Sparse coefficients are used as though they formed a full sequence

Pages 34–38 · Definition 5.1 and proof of Theorem C · arXiv:2605.30606v2

The theorem denotes by $a_i$ the nonzero coefficient attached to the sparse exponent $u_i$ , but its proof later writes sums over every exponent with the same symbol. Define the full sequence $\widetilde a_m=a_i$ when $m=u_i$ and $\widetilde a_m=0$ otherwise, and read those later sums with $\widetilde a_m$ . This makes the block decompositions and all displayed identities exact. The reference in Lemma 5.4 to the polynomial in Equation (3.7) should likewise point to Equation (4.9).

Theorem 6.5Typo

The periodic comparison index omits the preperiod offset

Page 38 · first paragraph of the proof of Theorem 6.5 · arXiv:2605.30606v2

For $U_n=a[0,r_n]$ and $V_n=a[r_n+1,s_n]$ , the digit of $U_nV_n^\infty$ at an index $m>r_n$ is $a_{r_n+1+((m-r_n-1)\bmod(s_n-r_n))}.$ The printed definition compares $a_{j-1}$ with $a_{(j-1)\bmod(s_n-r_n)}$ , omitting this offset. Insert the displayed shifted index in the definition of $j_n$ . It is the unique comparison consistent with $U_nV_n^\infty$ and restores the intended first-mismatch bounds, but it does not repair Lemma 6.4.

Theorem D · degree-one boundary and Remark 6.7Minor formal correction

The integer-base case and the alternative Schmidt citation need correction

Pages 40–41 · two proofs of Theorem D and Remark 6.7 · arXiv:2605.30606v2

In the $k$ -bonacci case the proof writes $[\mathbb Q(\beta):\mathbb Q]=2$ , although Theorem D allows an integer Pisot base of degree $1$ ; replace equality by $\leq2$ . Remark 6.7 cites W. M. Schmidt's paper on Northcott's theorem for a result actually due to K. Schmidt on eventually periodic greedy $\beta$ -expansions. More importantly, that theorem concerns the greedy expansion, whereas the coefficient words used here are not shown to be greedy-admissible. The remark therefore does not provide the claimed alternative proof without an additional admissibility or uniqueness argument. The main proof already obtains transcendence from the correctly applicable $k$ -bonacci and Sturmian sources.

K. Schmidt, On Periodic Expansions of Pisot Numbers and Salem Numbers ↗

Remaining proof chainCorrect and complete

Subspace-Theorem and quantitative arguments for Theorems A–C

Pages 18–38 · Sections 3–5 · arXiv:2605.30606v2

The normalized-place estimates, product-formula cancellation, quantitative Subspace-Theorem reduction, Siegel-lemma construction, Liouville inequalities, and Lenstra block separation are correctly matched to their hypotheses. After the explicit local notation corrections above, the dependencies proving Theorems A–C contain no further unsupported nontrivial step found in this audit.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2605.30606v2
Authors listed: Quang-Khai Nguyen
Audit date: August 15, 2026

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P(delta)

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P

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t>0

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Transcendence and measures via the refined Diophantine exponent

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Refined-Diophantine-exponent transcendence criterion

Quantitative transcendence measure and Mahler classification

Lacunary-series dichotomy and quantitative bounds

Exclusion of U1U_1U1​ and U2U_2U2​ for periodic-prefix approximants

The cited separation bound is false for arbitrary digit words

The height lower bound assumes two approximants are distinct

Ordinary equidistribution is not uniform for the shrinking intervals used

The chosen endpoint and mismatch bound require rounding and slack

The special linear form and auxiliary polynomials are evaluated at the wrong symbol

Sparse coefficients are used as though they formed a full sequence

The periodic comparison index omits the preperiod offset

The integer-base case and the alternative Schmidt citation need correction

Subspace-Theorem and quantitative arguments for Theorems A–C

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Exclusion of $U_1$ and $U_2$ for periodic-prefix approximants