arXiv:2601.10051v4

A note on exact approximations

Dmitry Gayfulin, Sergei Pitcyn

math.NT11J04

Paper metadata

Original submission: January 15, 2026
Latest revision: July 15, 2026 · version 4
Claimed MSC: 11J04
arXiv categories: math.NT
Paper license: arXiv non-exclusive distribution license ↗
Comments: 9 pages, more precise results and exposition

Abstract

Based on M. Hall's theorem we prove a simple result dealing with real numbers which admit exact approximations by rationals.

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Audit summary

Audited against arXiv v4

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

01Statements2 reported findingsCorrect

The two central exact-approximation theorems are supported under their stated ranges. The continued-fraction constructions produce the required Lagrange constant and control the approximation coefficient at every sufficiently large convergent.

Theorem 3Correct

Exact approximation from above for every $\gamma>5$

Pages 3–8 · Theorem 3 and Section 4 · arXiv:2601.10051v4

Perron's formula reduces the claim to constructing $\alpha$ with $\lambda_{n+1}(\alpha)>\gamma$ infinitely often and $\lambda_{n+1}(\alpha)<\gamma+\varpi(q_n)/q_n^2$ eventually. In each of the three ranges for $\gamma$ , the Hall-type decomposition supplies the two limiting tails. The marked central partial quotients make the corresponding Perron values converge to $\gamma$ from above, while every unmarked position stays uniformly below $\gamma$ . The current block lengths can then be chosen recursively: the preceding denominator is fixed before the new reversed block is lengthened, whereas $\varpi(q)$ tends to infinity. The continuant estimate therefore gives the required $o(q_n^{-2})$ one-sided excess at every marked position. These choices preserve all earlier requirements and establish both asserted rational-approximation conclusions.

Theorem 4Correct

Two-sided exact approximation in the stated Hall interval

Pages 3 and 8 · Theorem 4, Lemma 6, and its proof · arXiv:2601.10051v4

For $4+[0;3,1]+[0;2,1,3]\leq\gamma\leq5$ , the $F_3+F_3$ interval representation yields blocks whose marked Perron values approach $\gamma$ . Lemma 6 uniformly separates all unmarked values from $\gamma$ , and independently enlarging the reversed and forward portions of each block makes the two tail errors each smaller than $\varpi(q_{k_i-1})/(2q_{k_i-1}^2)$ . Hence the absolute error bound at the marked positions and the eventual upper bound elsewhere both follow. For $\gamma>5$ , the conclusion is the immediate weakening of Theorem 3 in the direction required by the larger upper approximation function.

02Proofs4 reported findingsCorrect

The proof reductions, block construction, parity choices, and recursive parameter selection are correct and complete after immediate-consequence closure. No substantive gap or literal notation defect was found.

Theorem 3 reductionCorrect and complete

Perron's formula and the convergent restriction

Pages 4–5 · Equations (16), (22), and (23) · arXiv:2601.10051v4

Legendre's criterion excludes nonconvergents from the stronger approximation range, and Perron's identity converts the two inequalities to the displayed bounds for the neighboring Perron values. Replacing the given increasing function by a smaller increasing function with $\varpi(q)/q^2<1/2$ is legitimate because proving finiteness for the larger resulting right-hand side implies finiteness for the original one. The shift between $n$ and $n+1$ is consistently handled by taking the convergent immediately preceding each marked digit.

Lemmas 1–5Correct and complete

Marked-block separation and recursive denominator control

Pages 5–8 · Lemmas 1–5, Corollary 4, and Equations (28)–(37) · arXiv:2601.10051v4

At marked digits, the forward and reversed tails converge to the two summands in the relevant Hall representation. At every other digit, either the leading partial quotient gives an immediate strict bound or the fixed separator block supplies a uniform neighborhood below $\gamma$ . The odd or even choices of block length give the stated continued-fraction order in Lemmas 2 and 5. The continuant inequality then transforms agreement of the first $m_i$ reversed digits into the quantitative estimate (34). Because the prefix denominator is fixed before $m_i$ is chosen, the recursive choice uses no circular dependence.

Lemma 6 and Theorem 4Correct and complete

Uniform control of noncentral positions

Page 8 · Lemma 6 and Equations (38)–(39) · arXiv:2601.10051v4

The explicit strict numerical gap below the left endpoint of the stated interval permits one fixed block threshold $N$ . Digits at most two have Perron value below $4$ , and a digit three is controlled by the two worst permitted neighboring tails. Thus only the marked fours can approach $\gamma$ . The two independently adjustable sides of each marked block then give (39), completing both the infinite-occurrence and eventual-exclusion assertions.

Hall-type decomposition inputsCorrect and complete

The cited interval statements are used with matching hypotheses

Pages 4 and 7–8 · Theorems 5–7 and Corollaries 1–3 · arXiv:2601.10051v4

The three ranges in the proof match the three stated sum-set intervals, including both shared endpoints. Only the existence of expansions with the displayed digit restrictions is used. Integer translation handles $\gamma>6$ , and the forbidden adjacent pairs in the intermediate range are used exactly to control positions carrying the digit four.

Astels, Cantor sets and numbers with restricted partial quotients ↗

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2601.10051v4
Authors listed: Dmitry Gayfulin, Sergei Pitcyn
Audit date: August 15, 2026

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A note on exact approximations

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Exact approximation from above for every γ>5\gamma>5γ>5

Two-sided exact approximation in the stated Hall interval

Perron's formula and the convergent restriction

Marked-block separation and recursive denominator control

Uniform control of noncentral positions

The cited interval statements are used with matching hypotheses

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Exact approximation from above for every $\gamma>5$