Proof-critical dependence graph

Diophantine approximation on abelian varieties; a conjecture of M. Waldschmidt

A statement-restricted recursive graph for the matrix reformulation of Waldschmidt's conjecture, the uniform inhomogeneous BA/VWA criteria, the abelian-variety exponent bound, the rank-one theorem, and ancillary isogeny invariance in Fishman–Lambert–Merrill–Simmons. The graph distinguishes externally rescued statements from defects in the posted proofs: a false local-minima lemma, a missing cutoff that makes the displayed exceptional set false, an invalid reverse winning-game proof, an isogeny-inequality sign error, and several repairable omissions.

Graph scope18 nodes22 proof-critical linksChecked September 26, 2026

Oriented proof graph

Dependence map

Arrows point from a prerequisite toward the paper whose marked statement uses it.

Solid arrow: headline proof lineage
Dashed arrow: a separately marked side or appendix claim
Dashed square: a terminal book

Hover over, or focus, a square to see its full citation.

Every visible arrow is documented in the evidence ledger below. A non-book leaf means that no earlier proof-critical source was identified for the marked statement—not that the paper has no other citations.

Evidence ledger

Proof-critical links

Density measure of rational points on Abelian varieties→Diophantine approximation on abelian varieties; a conjecture of M. WaldschmidtHeadline lineageSource link checked

Conjecture 1.1 is the reformulated target and Lemma 5.1 supplies the coefficient/height comparison. The quadratic relation between the two dummy scale parameters is exactly what changes the exponent from $\theta/2$ to $\theta$ .

Citation location: Waldschmidt Conjecture 1.1 and Lemma 5.1; focal Theorem 1.5 and Lemma 2.5Verification note: The exact source was checked; positive-definite height makes the two formulations equivalent modulo constants, torsion, and finite index.

Theorem 1.5Waldschmidt's conjecture for the abelian variety is equivalent to the logarithm matrix $J^{-1}H$ not being very well approximable.
Lemma 2.5The abelian-variety uniform approximation statement at height exponent $\theta/2$ is equivalent to the logarithm-matrix statement at coefficient exponent $\theta$ .

A metric theory of inhomogeneous Diophantine approximations with a fixed matrix→Diophantine approximation on abelian varieties; a conjecture of M. WaldschmidtHeadline lineageSource link partially checked

Exact v1 Theorems 1A–1B recover the uniform all-shift BA and non-VWA criteria. They supply the omitted proof of Theorem 2.1 and rescue Theorem 2.3 after the focal reverse winning-game proof fails.

Citation location: Moshchevitin–Neckrasov arXiv:2503.21180v1, Theorems 1A–1B; focal Theorems 2.1, 2.3 and Remark 2.4Verification note: The exact historical v1 statement was checked; the public audit is of the later v3.

Theorem 2.1A matrix $A$ is badly approximable exactly when one uniform Dirichlet constant works for every sufficiently large $Q$ and every shift $\gamma$ .
Theorem 2.3 $A$ is not very well approximable exactly when, for every $\varepsilon>0$ , a uniform all-shift inhomogeneous bound of exponent $n/m-\varepsilon$ holds for all sufficiently large $Q$ .

Logarithm laws for flows on homogeneous spaces→Diophantine approximation on abelian varieties; a conjecture of M. WaldschmidtHeadline lineageSource link partially checked

Lemma 8.3 and Theorem 8.5 provide the parametric-lattice bounds invoked in Lemma 2.2 and both directions of Theorem 2.3. They do not imply the false exact local-minima assertion in Lemma 2.2.

Citation location: Kleinbock–Margulis Lemma 8.3 and Theorem 8.5; focal Lemma 2.2 and Theorem 2.3Verification note: The source statements are valid, but the focal Lemma 2.2 inference is false; the bibliography also mislabels Lemma 8.3 as a theorem.

Lemma 2.2For a $\psi$ -approximable matrix, the local minima of $e^{r(t)}\lVert g_t\Lambda_A\rVert$ are asserted to occur exactly at the local minima of $\lVert g_t\Lambda_A\rVert$ .
Theorem 2.3 $A$ is not very well approximable exactly when, for every $\varepsilon>0$ , a uniform all-shift inhomogeneous bound of exponent $n/m-\varepsilon$ holds for all sufficiently large $Q$ .

Ein Übertragungsprinzip für konvexe Körper→Diophantine approximation on abelian varieties; a conjecture of M. WaldschmidtHeadline lineageSource link checked

Mahler's compound-body transference compares the successive minima needed to pass between the relevant lattice and dual-lattice estimates.

Citation location: Mahler §3; focal Theorem 2.3 proof

Theorem 2.3 $A$ is not very well approximable exactly when, for every $\varepsilon>0$ , a uniform all-shift inhomogeneous bound of exponent $n/m-\varepsilon$ holds for all sufficiently large $Q$ .

The set of badly approximable vectors is strongly $C^1$ incompressible→Diophantine approximation on abelian varieties; a conjecture of M. WaldschmidtHeadline lineageSource link partially checked

The focal reverse proof invokes HAW thickness and countable-intersection stability only to manufacture a counterexample shift. Those closure properties are correct, but the constructed target set is false as defined because its denominator cutoff is missing.

Citation location: BFKRW Proposition 2.3(a)–(b); focal Theorem 2.3Verification note: The imported HAW facts are correct; their focal application does not repair the invalid reverse proof.

Theorem 2.3 $A$ is not very well approximable exactly when, for every $\varepsilon>0$ , a uniform all-shift inhomogeneous bound of exponent $n/m-\varepsilon$ holds for all sufficiently large $Q$ .

An Introduction to Diophantine Approximation→Diophantine approximation on abelian varieties; a conjecture of M. WaldschmidtHeadline lineageSource link partially checked

The proof opens by transferring non-VWA from $A$ to $A^T$ . This material step is uncited in the focal paper; classical transpose transference supplies it.

Citation location: Cassels Chapter V, §2; uncited opening step of focal Theorem 2.3Verification note: The classical source family is identified, but the focal paper gives no pinpoint and this audit does not claim a newly checked book scan.

Theorem 2.3 $A$ is not very well approximable exactly when, for every $\varepsilon>0$ , a uniform all-shift inhomogeneous bound of exponent $n/m-\varepsilon$ holds for all sufficiently large $Q$ .

On shrinking targets for $\mathbb{Z}^m$ actions on tori→Diophantine approximation on abelian varieties; a conjecture of M. WaldschmidtHeadline lineageSource link checked

Theorem 1 gives a nonempty full-dimensional set of uniformly inhomogeneously badly approximable shifts, contradicting any exponent above $r/(2g)$ that held for every target.

Citation location: BHKV Theorem 1; focal Lemma 2.6Verification note: Exact arXiv v2 has the correct KTV child-count sign and proves the selected full-dimension statement in the ball setting.

Lemma 2.6For a simple abelian variety of dimension $g$ and Mordell–Weil rank $r$ , $\sigma_{\mathcal A,K}\leq r/(2g)$ .

Minkowski's theorem on nonhomogeneous approximation→Diophantine approximation on abelian varieties; a conjecture of M. WaldschmidtHeadline lineageSource chain not fully resolved

Niven's nonhomogeneous $1/4$ theorem is the sharp circle-rotation estimate used in the elliptic rank-one bound.

Citation location: Niven, Proc. AMS 12 (1961), 992–993; focal Theorem 3.1Verification note: Bibliography and statement are corroborated, but the exact original proof was blocked by AMS access controls.

Theorem 3.1For a rank-one elliptic curve, $\sigma_{E,K}=1/2$ ; the proof also states a uniform weak Dirichlet estimate and its optimality.

The Arithmetic of Elliptic Curves→Diophantine approximation on abelian varieties; a conjecture of M. WaldschmidtHeadline lineageSource-chain endpoint

Elliptic uniformization and canonical-height comparison identify the group orbit and convert Niven's rotation estimate to the focal height-normalized bound.

Citation location: Silverman Chapters VI and VIII; focal Section 3

Theorem 3.1For a rank-one elliptic curve, $\sigma_{E,K}=1/2$ ; the proof also states a uniform weak Dirichlet estimate and its optimality.
Lemma 3.2The elliptic-curve approximation expression is reduced to an inhomogeneous rotation expression.

The Arithmetic of Elliptic Curves→Diophantine approximation on abelian varieties; a conjecture of M. WaldschmidtAncillary claimSource chain not fully resolved

The focal proof cites Silverman for height functoriality under an isogeny of arbitrary abelian varieties, but the cited book treats elliptic curves; a general abelian-variety source is needed.

Citation location: Focal Lemma 1.7 citation to Silverman Chapter VIII, §5Verification note: Citation-scope mismatch; the mathematical claim is standard but this source does not justify the stated generality.

Lemma 1.7The Diophantine exponent is invariant under every nonconstant $K$ -morphism (isogeny) of abelian varieties.

Positivity in Algebraic Geometry I: Classical Setting: Line Bundles and Linear Series→Diophantine approximation on abelian varieties; a conjecture of M. WaldschmidtAncillary claimSource-chain endpoint

The dual isogeny gives the reverse implication once one direction of isogeny invariance is established.

Citation location: Lazarsfeld Theorem 1.2.13 as cited; focal Lemma 1.7

Lemma 1.7The Diophantine exponent is invariant under every nonconstant $K$ -morphism (isogeny) of abelian varieties.

Heights in Diophantine Geometry→Diophantine approximation on abelian varieties; a conjecture of M. WaldschmidtAncillary claimSource chain not fully resolved

The focal proof cites this book for metric comparison under isogeny, although the exact locator was not verified and the needed compact-manifold comparison is elementary.

Citation location: Focal Lemma 1.7 citation to Chapter 8, §8.2, item 27

Lemma 1.7The Diophantine exponent is invariant under every nonconstant $K$ -morphism (isogeny) of abelian varieties.

Approximation simultanée par des produits de puissances de nombres algébriques→Density measure of rational points on Abelian varietiesHeadline lineageSource link checked

Waldschmidt 1999 extends the simultaneous approximation estimate of Lemma 3.1 from the 1997 paper to the abelian-variety setting.

Citation location: Waldschmidt 1999, proof of Lemma 5.1; Waldschmidt 1997, Lemma 3.1

Lemma 5.1Relates a coefficient box of radius $T$ to canonical height of order $T^2$ .

An Introduction to Diophantine Approximation→Approximation simultanée par des produits de puissances de nombres algébriquesHeadline lineageSource-chain endpoint

Waldschmidt's Lemma 3.2 is explicitly identified with Cassels' Theorem XVII and its two directions are applied in the proof of Lemma 3.1.

Citation location: Cassels Chapter V, §8, Theorem XVII; Waldschmidt 1997, Lemmas 3.1–3.2

Lemma 3.1The simultaneous approximation/transference estimate imported into Waldschmidt's 1999 lemma.

Exponents of homogeneous and inhomogeneous Diophantine approximation→A metric theory of inhomogeneous Diophantine approximations with a fixed matrixHeadline lineageSource link partially checked

Bugeaud–Laurent Lemma 3 supplies the transference inequality used in the v1 lower-bound branch of the fixed-matrix dichotomy.

Citation location: Bugeaud–Laurent Lemma 3; Moshchevitin–Neckrasov v1 proof of Theorem 1AVerification note: Proof-bearing arXiv counterpart checked rather than a publisher file.

Theorems 1A–1B (v1)The v1 fixed-matrix transference results recover both the uniform all-shift BA criterion and the uniform all-shift non-VWA exponent criterion used by the focal paper.

Metric Theory of Diophantine Approximations→A metric theory of inhomogeneous Diophantine approximations with a fixed matrixHeadline lineageSource-chain endpoint

The cited second-moment lemma turns the v1 correlation estimates into the almost-everywhere divergent-branch conclusion.

Citation location: Sprindžuk Chapter 1, Lemma 5 as cited; Moshchevitin–Neckrasov v1 proof of Theorem 1B

Theorems 1A–1B (v1)The v1 fixed-matrix transference results recover both the uniform all-shift BA criterion and the uniform all-shift non-VWA exponent criterion used by the focal paper.

Geometrie der Zahlen→Ein Übertragungsprinzip für konvexe KörperHeadline lineageSource-chain endpoint

Mahler's Section 3 proof applies Minkowski's successive-minima product inequalities to the body and its compound.

Citation location: Mahler §3; Minkowski, Geometrie der Zahlen

Section 3 transference theoremCompares successive minima of a convex body and its compound body.

Diophantine approximation and badly approximable sets→On shrinking targets for $\mathbb{Z}^m$ actions on toriHeadline lineageSource link checked

KTV Theorem 1 is the full-dimension criterion and Lemma 7 supplies the local counting estimate. The original theorem has the required lower bound that BHKV accidentally prints with the opposite sign.

Citation location: KTV Theorem 1 and Lemma 7; BHKV proof of Theorem 1Verification note: The selected BHKV ball-case use is unaffected by the broader product-coordinate repair recorded on the KTV node, and arXiv v2 reproduces the correct lower cardinality bound.

Theorem 1A lacunary toral shrinking-target orbit has full Hausdorff dimension for every target.

Exponents of homogeneous and inhomogeneous Diophantine approximation→On shrinking targets for $\mathbb{Z}^m$ actions on toriHeadline lineageSource link partially checked

Section 5 supplies the lacunary best-approximation subsequence around which BHKV builds its resonant sets.

Citation location: Bugeaud–Laurent §5, especially equation (13); BHKV proof of Theorem 1Verification note: Proof-bearing arXiv counterpart checked rather than a publisher file.

Theorem 1A lacunary toral shrinking-target orbit has full Hausdorff dimension for every target.

An Introduction to Diophantine Approximation→On shrinking targets for $\mathbb{Z}^m$ actions on toriHeadline lineageSource-chain endpoint

The linear-forms version of Minkowski's theorem gives the integer-vector existence estimate used in the BHKV construction.

Citation location: Cassels Appendix B; BHKV proof of Theorem 1

Theorem 1A lacunary toral shrinking-target orbit has full Hausdorff dimension for every target.

Fractal Geometry: Mathematical Foundations and Applications→Diophantine approximation and badly approximable setsHeadline lineageSource-chain endpoint

KTV invokes the mass distribution principle to convert its Cantor measure estimates into the final Hausdorff-dimension lower bound.

Citation location: Falconer p. 55; KTV proof of Theorem 1

Theorem 1 and Lemma 7The general full-dimension criterion and local counting estimate used by BHKV.

An Introduction to Diophantine Approximation→Exponents of homogeneous and inhomogeneous Diophantine approximationHeadline lineageSource-chain endpoint

Bugeaud–Laurent explicitly derives Lemma 3 from Part B of Cassels' Theorem XVII.

Citation location: Cassels Chapter V, §8, Theorem XVII(B); Bugeaud–Laurent Lemma 3

Lemma 3Homogeneous transference inequality used in the Moshchevitin–Neckrasov lower-bound branch.