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arXiv:1808.05845v2

Popular Products and Continued Fractions

Nikolay Moshchevitin, Brendan Murphy, Ilya Shkredov

math.NTmath.CO11A5511B30
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Paper metadata

Original submission
August 17, 2018
Latest revision
August 23, 2018 · version 2
Claimed MSC
11A55, 11B30
arXiv categories
math.NT, math.CO

Abstract

We prove bounds for the popularity of products of sets with weak additive structure, and use these bounds to prove results about continued fractions. Namely, we obtain a nearly sharp upper bound for the cardinality of Zaremba's set modulo pp.

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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 20, 2026
01Statements2 reported findingsCorrect

The reciprocal-sum expansion bounds, popular-product corollaries, and the near-optimal upper bound for Zaremba's set modulo a prime are correct.

Theorems 1, 4, and 6Correct

The rich-transformation bounds yield the stated continued-fraction estimate

Pages 3–9 and 15–30 · principal theorems and their proofs · arXiv:1808.05845v2

Small additive expansion makes every associated linear fractional transformation rich on one common set. The flattening and subgroup non-concentration estimates bound that richness, giving the reciprocal-sum alternatives. Applying the interval version to the two halves of a bounded-partial-quotient expansion produces the exponent 2wM−1+ε(1−wM)2w_M-1+\varepsilon(1-w_M).

Full paper, version 2 ↗
Adjoint notationTypo · no status impact

The multiplicative-group adjoint is printed with an additive minus sign

Pages 17 and 26 · definition of the adjoint and proof of Theorem 24 · arXiv:1808.05845v2

For a function on SL⁡2(Fp)\operatorname{SL}_2(\mathbb F_p) the adjoint is μ~(x)=μ(x−1)\widetilde\mu(x)=\mu(x^{-1}), not μ(−x)\mu(-x). The convolution identities later in the paper, including the correctly printed definition on page 21, all use inversion. Replacing the two minus signs by inverses restores consistent notation.

02Proofs2 reported findingsCorrect

The incidence, flattening, girth, and continued-fraction proof chains are correct and complete after the adjoint notation typo is repaired.

Proof of Theorem 6Correct and complete

The bounded-partial-quotient decomposition gives the announced exponent

Pages 7–9 · Lemmas 7–8 and proof of Theorem 6 · arXiv:1808.05845v2

Splitting a finite continued fraction at a denominator near pβp^\beta places its two halves in the sets AβA_\beta and BβB_\beta, whose cardinalities follow from Hensley's exponent wMw_M. Theorem 4 bounds their additive covering, and optimizing β\beta yields ∣A∣≪p2wM−1+ε(1−wM)|A|\ll p^{2w_M-1+\varepsilon(1-w_M)} with the stated choice of MM.

Sections 4–8 and appendicesCorrect and complete after the stated repair

The expansion estimates are applied with valid subgroup and girth hypotheses

Pages 10–37 · rich transformations, flattening, and appendices · arXiv:1808.05845v2

Dickson's subgroup classification gives the required coset non-concentration, the locally free walk estimate supplies the flattening parameter, and the doubly transitive action yields the incidence bound. The special matrix generators have the claimed girth, so Theorem 11 and hence Theorems 4 and 6 follow.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:1808.05845v2
Authors listed
Nikolay Moshchevitin, Brendan Murphy, Ilya Shkredov
Audit date
August 20, 2026
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