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Paper metadata

Original submission
July 9, 2026
Claimed MSC
11H46, 11J20
arXiv categories
math.NT
Comments
13 pages

Abstract

We show that a fully inhomogeneous uniform Littlewood type problem has a negative answer and the counterexamples form a dense GδG_δ set. This extends the author's recent analogous result for the homogeneous case. The main difficulty in the general setting is the semi-homogeneous case where one linear form is homogeneous and the other inhomogeneous with irrational shift. We further address the higher dimensional analogue.

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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.

Current report

Detailed mathematical audit

Generated August 15, 2026
01Statements2 reported findingsContains unsupported statements

The two-dimensional inhomogeneous uniform Littlewood results are supported. The higher-dimensional Theorem 3.1 is not able to be verified because its stated simultaneous-approximation hypothesis supplies neither the coordinatewise lower bounds nor the coprimality required by its proof.

Theorems 1.2 and 2.1Correct

Dense counterexample sets in the two-dimensional cases

Pages 2–9 · Theorems 1.2 and 2.1 and Sections 2.1–2.3 · arXiv:2607.08114v1

The homogeneous and rational-shift cases reduce to the stated known inputs. In the fully inhomogeneous case, rational parameter pairs are dense and make one factor uniformly separated from zero along every denominator. In the semihomogeneous case, adjacent convergents and the modular-hyperbola count provide nested intervals with a uniform positive lower bound for the product. Intersecting the resulting open dense sets gives exactly the asserted dense GδG_\delta conclusions.

Theorem 3.1Not able to verify

The higher-dimensional conclusion is not established by the stated hypothesis

Pages 10–12 · Theorem 3.1, Lemma 3.2, and its proof · arXiv:2607.08114v1

Condition (16) gives only γ1q−(k−1)<max⁡j∥qθj∥<γ2q−(k−1)\gamma_1q^{-(k-1)}<\max_j\|q\theta_j\|<\gamma_2q^{-(k-1)} for infinitely many qq. The construction later needs a lower bound for each coordinate in order to impose condition (C3)(C3) for every jj; a lower bound on the maximum controls only one coordinate. Moreover, Lemma 3.2 explicitly assumes (R1⋯Rk−1,q)=1(R_1\cdots R_{k-1},q)=1, while Theorem 3.1 does not ensure that the nearest integers RjR_j have this coprimality. No counterexample to the theorem was found, but the stated assumptions and supplied proof do not resolve these two obligations.

02Proofs3 reported findingsContains incorrect or incomplete proofs

The two-dimensional constructions are correct and complete. The proof of Theorem 3.1 applies its counting lemma without establishing two material hypotheses, so the higher-dimensional argument is incomplete as written.

Sections 2.1–2.3Correct and complete

Rational, fully inhomogeneous, and semihomogeneous cases

Pages 3–9 · proof of Theorem 1.2 · arXiv:2607.08114v1

The rational-shift reduction preserves the required uniform quantifier. The dense rational parameter set handles the case in which neither form is homogeneous. For the remaining semihomogeneous case, Proposition 2 gives the necessary adjacent-convergent separation and Lemma 2.2 provides enough residue classes in each interval; the inductive nesting therefore produces a dense set with a uniform positive product bound.

Proof of Theorem 3.1Incomplete as written

Coordinatewise separation and coprimality are missing

Pages 10–12 · condition (16), Lemma 3.2, and proof of Theorem 3.1 · arXiv:2607.08114v1

The proof uses condition (C3)(C3) separately for all jj, but the lower half of (16) bounds only the largest value of ∥qθj∥\|q\theta_j\|. It also invokes Lemma 3.2 with residues RjR_j although the lemma requires (R1⋯Rk−1,q)=1(R_1\cdots R_{k-1},q)=1 and the theorem supplies no such condition. Repair classification: Plausible repair only. Stronger hypotheses giving coordinatewise two-sided bounds and coprime approximating numerators would make the printed counting step applicable, but that would change the theorem's scope; alternatively, a new simultaneous-selection argument is needed.

End of Section 3Typo

The proof refers to the wrong theorem number

Page 11 · opening of the final proof · arXiv:2607.08114v1

The phrase “proof of Theorem 3.2” must read “proof of Theorem 3.1”; there is no Theorem 3.2, and the surrounding section is explicitly proving Theorem 3.1. The correction is unique and has no mathematical effect.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:2607.08114v1
Authors listed
Johannes Schleischitz
Audit date
August 15, 2026
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