arXiv:0903.5537v1

Differentiability of the Minkowski question mark function

Anna A. Dushistova, Igor D. Kan, Nikolai G. Moshchevitin

math.NT11J7011A55

Paper metadata

Original submission: March 31, 2009
Claimed MSC: 11J70, 11A55
arXiv categories: math.NT
Paper license: arXiv non-exclusive distribution license ↗
Comments: 38 pages

Abstract

We prove new results on the derivative of the Minkowski question mark function. Some of our theorems are non-improvable.

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Audit summary

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.

Current report

Detailed mathematical audit

Generated August 20, 2026

01Statements4 reported findingsContains unsupported statements

The derivative criteria and constructions in Theorems 1–4, the first part of Theorem 5, the second part of Theorem 6, and Theorem 7 are supported. Theorem 6(i) is not verified because a counting invariant used essentially in its proof is false as printed; Theorem 5(ii) inherits that unresolved dependency. No counterexample to either statement was established.

Theorems 1–4Correct

Unbounded-partial-quotient derivative thresholds and sharp constructions

Pages 2–3 and 25–33 · Theorems 1–4 and their proofs · arXiv:0903.5537v1

The continued-fraction derivative criterion reduces the assertions to upper or lower bounds for continuants. Lemmas 5–15 supply the extremal continuant estimates, and the block constructions in the converse directions produce the required oscillating or one-sided quotient behavior. The two uniquely determined notation corrections in the proof of Theorem 1(ii), recorded below, do not alter its construction or conclusion.

Theorems 5(i), 6(ii), and 7Correct

Verified bounded-partial-quotient results

Pages 4–5 and 33 and 36–38 · Theorems 5(i), 6(ii), and 7 · arXiv:0903.5537v1

For Theorem 5(i), the two cases for the auxiliary growth function are exhaustive: the unbounded case invokes Lemmas 1 and 15, while the bounded case forces an eventually all-one continued fraction. The explicit alternating-block construction for Theorem 6(ii) gives a vanishing derivative and the stated $15n\sqrt{t}$ bound. For $E_4$ , the continuant transformations in the proof of Theorem 7 yield a uniform base strictly larger than $2$ and hence the asserted infinite derivative.

Theorem 6(i)Not able to verify

The quantitative lower-deviation claim is not verified from the exact preprint

Pages 5 and 34–36 · Theorem 6(i) and its sketched proof · arXiv:0903.5537v1

The proof's bound on the number of unit variations rests on the stated decrease of $\sigma_1^{(n)}$ , but that decrease is formally false for the printed definition, as shown below. Two additional continuant estimates on which the final constant depends are asserted without derivation. These defects do not disprove the theorem, but no correction preserving the claimed constant was verified for this exact version.

Theorem 5(ii)Not able to verify

The slow-growth reduction is valid, but the derivative conclusion depends on Theorem 6(i)

Pages 4 and 33 · Theorem 5(ii) and its proof · arXiv:0903.5537v1

There is no gap in passing from an arbitrary increasing unbounded $\psi$ to the case $\psi(t)=o(\sqrt{t})$ : one may perform the construction with $\phi(t)=\min\{\psi(t),t^{1/4}\}$ , which is increasing, unbounded, bounded above by $\psi$ , and $o(\sqrt{t})$ . The unresolved point is different. The proof uses Theorem 6(i) to exclude a zero derivative for the constructed number, so the unverified proof of Theorem 6(i) leaves this statement unsupported in the exact preprint.

02Proofs6 reported findingsContains incorrect or incomplete proofs

Most proof chains close after two mechanical notation corrections. The proof of Theorem 6(i), however, uses a variation count contradicted by its own definition and omits two nontrivial continuant estimates; Theorem 5(ii) depends on that result.

Proof of Theorem 1(ii)Typo

Two indices in the block recursion have uniquely determined corrections

Pages 26–27 · proof of Theorem 1(ii), conditions (c0)–(c3) · arXiv:0903.5537v1

The displayed definition of $b_{k+1}$ must subtract $\sum_{j=0}^{k}c_j$ , not $\sum_{j=1}^{k}c_j$ ; otherwise the initial block of $c_0>0$ ones is omitted and the following bound for $S_x(t_{k+1})$ fails by exactly $c_0$ . Also, condition (c1) must contain $\psi(t_k+c_k+1)$ , not $\psi(t_k+c+k+1)$ . The surrounding recursion and the identities $t_{k+1}=t_k+c_k+1$ and $Q_k=q_{t_{k+1}}$ uniquely determine both corrections. They are notation-level errors and do not lower the statement status.

Proofs of Theorems 1–4Correct and complete

The continuant estimates and case divisions close

Pages 25–33 · Sections 11.1–11.8 · arXiv:0903.5537v1

The direct parts combine the derivative lemma with the relevant lower continuant bounds. The converse block constructions alternate long extremal blocks with sparse corrective digits, and their endpoint estimates extend to every intermediate index by monotonicity. In Theorem 4(i), the two alternatives supplied by the unit-variation procedure both force the displayed square-root deviation; the construction in Theorem 4(ii) gives the matching upper order. No omitted case affecting these conclusions was found.

Proof of Theorem 5(ii)Correct reduction

The arbitrary-growth-function reduction is an immediate valid consequence

Page 33 · Section 11.10 · arXiv:0903.5537v1

Although the text writes out the construction only under $\psi(t)=o(\sqrt{t})$ , replacing the requested function by $\min\{\psi(t),t^{1/4}\}$ gives that hypothesis and a stronger displayed bound. No correction to the theorem statement is needed. The only remaining proof obligation is the separate dependence on Theorem 6(i).

Proof of Theorem 6(i)Incorrect as written · no verified repair supplied

The claimed decrease of the variation invariant is false

Page 34 · Section 11.11, definition of $\sigma_1^{(n)}$ and the variation algorithm · arXiv:0903.5537v1

With the printed conventions $a_{i_0}=1$ and $a_{i_{k+1}}=n$ , the definition telescopes to $\sigma_1^{(n)}=\sum_{j=0}^{k}\bigl(n-1-(a_{i_{j+1}}-a_{i_j})\bigr)=k(n-1).$ Consequently, moving the smallest interior digit down by one and the largest interior digit up by one leaves $\sigma_1^{(n)}$ unchanged whenever both remain interior. For example, with $n=10$ and interior digits $4,6$ , the replacement $4,6\mapsto3,7$ leaves $k=2$ and hence leaves $\sigma_1^{(10)}=18$ , contradicting the claim that every step decreases it by at least $4$ . The ensuing bound that the first stage takes at most $\sigma_1^{(n)}/4$ steps is therefore invalid. No uniquely determined replacement invariant was recoverable from the preprint.

Theorem 6(i) continuant estimatesIncomplete as written · no verified repair supplied

Two nontrivial estimates are only asserted by analogy

Pages 34–35 · Equations (97) and (99) in Section 11.11 · arXiv:0903.5537v1

Equation (97) is the new bounded-digit analogue controlling the continuant by $\sigma_1^{(n)}$ , and Equation (99) combines it with the permutation statistic used in the rest of the proof. Both are asserted as analogous to earlier formulas without a derivation. They are not routine substitutions: the constants and the later dichotomy depend on the precise number and effect of the new unit variations. Together with the false variation count, this prevents verification of the proof's final quantitative constant.

Proofs of Theorem 6(ii) and Theorem 7Correct and complete

The independent bounded-digit arguments remain valid

Pages 36–38 · Sections 11.12–11.13 · arXiv:0903.5537v1

Theorem 6(ii) uses a direct alternating-block construction and does not invoke the defective invariant from part (i). Its recurrence gives $O(nk)$ deviation while $t_k\gg k^2$ . Theorem 7 independently reduces to extremal digits and compares the resulting continuant with $2^{S_x(t)/2}$ ; the strict exponential gain proves the derivative is infinite throughout $E_4$ .

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:0903.5537v1
Authors listed: Anna A. Dushistova, Igor D. Kan, Nikolai G. Moshchevitin
Audit date: August 20, 2026

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Differentiability of the Minkowski question mark function

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Unbounded-partial-quotient derivative thresholds and sharp constructions

Verified bounded-partial-quotient results

The quantitative lower-deviation claim is not verified from the exact preprint

The slow-growth reduction is valid, but the derivative conclusion depends on Theorem 6(i)

Two indices in the block recursion have uniquely determined corrections

The continuant estimates and case divisions close

The arbitrary-growth-function reduction is an immediate valid consequence

The claimed decrease of the variation invariant is false

Two nontrivial estimates are only asserted by analogy

The independent bounded-digit arguments remain valid

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