Published paper
On exponents of homogeneous and inhomogeneous Diophantine approximation
MSC 2020
11J2011J1311J82
Abstract
The paper develops homogeneous-to-inhomogeneous transference through Diophantine exponents. Its best-approximation growth and transference lemmas are used in the fixed-matrix dependence graph.
Role in dependence graphs
Proof-critical source
Metric theory with a fixed matrix
This paper is included only for the following marked statements:
- Lemma 1 · §2, immediately after equations (4)–(5)A sequence of best-approximation denominators grows at least geometrically. Bugeaud–Laurent give the full pigeonhole proof.
- Lemma 3 · §3, Transference lemmaTurns a uniform lower bound for the transpose into an inhomogeneous approximation for every shift. Its reproduced proof invokes Cassels, Chapter V, Theorem XVII, Part B.
Proof-critical source
Diophantine approximation on abelian varieties; a conjecture of M. Waldschmidt
This paper is included only for the following marked statements:
- Lemma 3 · arXiv counterpart, Section 3Homogeneous transference inequality used in the Moshchevitin–Neckrasov lower-bound branch.
- Section 5 lacunary subsequence · arXiv counterpart, Section 5, equation (13)Constructs the lacunary best-approximation sequence used in BHKV.
Proof-critical source
Twisted Diophantine Approximation II: Uniform Theory
This paper is included only for the following marked statement:
- Lemma 3 · §3, Transference lemmaThe proof-bearing formulation from which Moshchevitin--Neckrasov recall Lemma I.
AI-generated audit
Audit summary
Not a correctness certificate. These reports do not replace expert scrutiny or formal verification.
Exact reviewed source
Frozen exact arXiv:math/0406065v1 · fallback for the inaccessible version of record
Yann Bugeaud and Michel Laurent. Exponents of inhomogeneous Diophantine approximation. arXiv:math/0406065v1; journal version: On exponents of homogeneous and inhomogeneous Diophantine approximation, Moscow Mathematical Journal 5 (2005), 747–766.
The exact Moscow Mathematical Journal version of record was not anonymously retrievable. This report therefore evaluates the authors' exact arXiv v1 manuscript and does not claim to audit the version of record.
Open audited source ↗01Statements2 reported findingsNo statement issues found
The reciprocal transference formulas for generic inhomogeneous exponents, the universal lower bounds, the explicit badly approximable shift, and the Hausdorff-dimension drop are correct. No counterexample or quantifier mismatch was found.
Main theoremNo issue found
Generic inhomogeneous exponents are the reciprocals of the dual homogeneous exponents
arXiv:math/0406065v1 · §1 theorem; proof in §4
The transference lemma gives and for every shift. For almost every shift, Borel–Cantelli supplies lower bounds for the scalar products with the best-approximation vectors. The duality identity then gives the reverse inequalities after letting the two auxiliary exponents approach their endpoints. The two limsup/uniform quantifiers are handled separately and in the correct order.
Exact arXiv v1 manuscript ↗Propositions 1 and 2No issue found
The existence and exceptional-dimension conclusions follow with the stated thresholds
arXiv:math/0406065v1 · §§1, 4, and 6
A lacunary subsequence of best approximations and Cassels' avoidance lemma give the explicit shift in Proposition 1. For Proposition 2, the exceptional scalar-product strips admit a summable Hausdorff cover of every dimension above ; outside that limsup set the main duality calculation gives a strict upper bound below the prescribed . This proves both nullity and the strict dimension drop.
02Proofs3 reported findingsNo proof issues found
The best-approximation growth, Borel–Cantelli, transference, and Hausdorff-cover arguments are coherent. One bounded-solution case is left implicit in the universal lower-bound proof, but the standard reduction to homogeneous approximation closes it without changing the theorem.
Lemma 1No issue found in the reviewed proof
Best-approximation denominators grow at least geometrically
arXiv:math/0406065v1 · §2, Lemma 1
Among consecutive approximation vectors, two agree modulo in all integer and nearest-integer coordinates. Their divided difference is a nonzero integer vector with strictly smaller error, forcing . Iteration gives the asserted exponential lower bound and exactly the summability needed in Lemma 2.
Lemma 3Correct conditional on cited terminal theorem
The inhomogeneous transference step correctly isolates its external input
arXiv:math/0406065v1 · §3, Lemma 3
Cassels' Chapter V, Theorem XVII(B) reduces solvability to inequality (7). With , large dual vectors are controlled by the term, short nonzero vectors by the assumed lower bound on , and trivially. Thus every case required by the cited theorem is covered.
Universal lower-bound proofSuspected minor formal issue
Unbounded approximants are implicit rather than stated
arXiv:math/0406065v1 · beginning of §4
Applying Lemma 3 at arbitrarily large scales produces integer solutions but the text immediately treats their norms as unbounded. If a bounded subsequence occurs, one fixed vector gives an exact inhomogeneous congruence. Translating by that vector reduces the claim to the homogeneous problem, whose required exponent follows from the classical transference bound recalled in §1. Adding this two-case sentence makes the deduction formally complete.
03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.