arXiv:2506.19060v1

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

Lior Fishman, David Lambert, Keith Merrill, David Simmons

math.NT11J20

Paper metadata

Original submission: June 23, 2025
Claimed MSC: 11J20
arXiv categories: math.NT
Paper license: CC BY 4.0 ↗
Comments: 11 pages

Abstract

Following the work of Waldschmidt, we investigate problems in Diophantine approximation on abelian varieties. First we show that a conjecture of Waldschmidt for a given simple abelian variety is equivalent to a well-known Diophantine condition holding for a certain matrix related to that variety. We then posit a related but weaker conjecture, and establish the upper bound direction of that conjecture in full generality. For rank 1 elliptic curves defined over a number field $K \subset \mathbb{R}$ , we then obtain a weak-type Dirichlet theorem in this setting, establish the optimality of this statement, and prove our conjecture in this case.

Dependence graphs

Proof lineage

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.

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AI-generated audit

Audit summary

Audited against arXiv v1

Not a correctness certificate. A green “no issues found” result may include yellow suspected typos or minor formal issues 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 September 26, 2026

01Statements7 reported findingsSuspected incorrect statements

The headline equivalence with non-very-well-approximability, the general upper bound, and the rank-one elliptic-curve exponent appear mathematically correct after the explicit repairs recorded below; Theorem 2.3 is also independently supplied by the cited Moshchevitin-Neckrasov theorem. The manuscript nevertheless contains false auxiliary statements: Lemma 2.2 fails for matrices with an exact integer relation, the set declared hyperplane absolute winning on page 7 omits its denominator cutoff and can be empty, and the proof of isogeny invariance states the comparison inequality in the wrong direction. The rank-zero case of Theorem 1.5 is also not defined under the paper's matrix conventions.

Theorems 1.5 and 2.3Correct after the stated scope and proof repairs

The Waldschmidt equivalence is correct in positive rank

PDF pages 3 and 6-8 · Theorems 1.5 and 2.3, Lemma 2.5, and Remark 2.4 · arXiv:2506.19060v1

For rank $r>0$ , Waldschmidt's Lemma 5.1 converts the height exponent to the coefficient exponent by $\widehat h(\sum q_j\beta_j)\asymp\|q\|^2$ . With $B=J^{-1}H$ , Theorem 2.3 applied with $m=g$ and $n=r$ then says exactly that the uniform error $Q^{-r/g+\varepsilon}$ holds for every shift if and only if $B$ is not very well approximable. The finite-index passage from all rational points to $\Gamma=\sum_j\mathbb Z\beta_j$ , and invariance under changing periods, lifts, or a full-rank rational subgroup, are standard but should be stated. The reverse proof printed in this paper is invalid; the result itself is independently established by Theorem 1(A)-(B) of arXiv:2503.21180v1.

Moshchevitin and Neckrasov, exact arXiv:2503.21180v1 ↗

Lemma 2.2Suspected incorrect statement as written

The local-minimum statement fails when there is an exact relation

PDF page 5 · Lemma 2.2 and proof · arXiv:2506.19060v1

Take $m=n=1$ , $A=0$ , and $\psi(x)=1/x$ . The matrix is $\psi$ -approximable, because $Aq+p=0$ for every nonzero $q$ with $p=0$ . But $\Lambda_A=\mathbb Z^2$ and, for $t\geq0$ , $\|g_t\Lambda_A\|=e^{-t}$ , which is strictly decreasing and has no local minima. Thus there is no unbounded sequence $\{t_k\}$ of the asserted kind. The lemma becomes plausible after excluding a nonzero lattice vector whose expanding block vanishes and treating that exact-relation case separately; the paper does neither, and later invokes the lemma in a case that includes rational very-well-approximable matrices.

Exact manuscript, arXiv:2506.19060v1 ↗

Theorem 2.3 reverse directionSuspected incorrect statement as written

The displayed target set is not hyperplane absolute winning as defined

PDF page 7 · definition of $D_{c,\mu}$ and the claimed HAW strengthening · arXiv:2506.19060v1

The definition of $D_{c,\mu}$ quantifies over all $q\in\mathbb Z^n$ but omits the required restriction $\|q\|<Q$ . For $m=n=1$ and an irrational Liouville number $A$ , the orbit $A\mathbb Z+\mathbb Z$ is dense, so for every $\gamma$ and every positive threshold there is an unrestricted $q$ below that threshold. Hence the printed $D_{c,\mu}$ is empty, and therefore cannot be HAW. Restoring $\|q\|<Q$ is necessary, but the subsequent proof also needs the repairs recorded in the proofs panel.

Exact manuscript, arXiv:2506.19060v1 ↗

Lemma 1.7Suspected incorrect as written · an apparently sound repair

The comparison inequality is reversed although isogeny invariance survives

PDF page 4 · proof of Lemma 1.7 · arXiv:2506.19060v1

Let $P_2\in A_2(\mathbb R)_0\setminus A_2(K)$ and choose a real preimage $P_1$ under the isogeny. A rational preimage would map to a rational point, so $P_1\notin A_1(K)$ . Mapping approximants to $P_1$ shows $\sigma(P_2)\geq\sigma(P_1)$ and hence $\sigma_{A_2,K}\geq\sigma_{A_1,K}$ , not the displayed $\leq$ . Applying the same corrected inequality to the dual isogeny gives the reverse inequality and restores the stated equality. Thus the lemma's conclusion is sound, but the only inequality actually stated in its proof is false.

Exact manuscript, arXiv:2506.19060v1 ↗

Theorem 1.5 at rank zeroSuspected ill-defined boundary case

The matrix characterization has an unstated positive-rank hypothesis

PDF pages 2-3 and 8 · Definition 1.3, Theorem 1.5, and its proof · arXiv:2506.19060v1

Definition 1.3 and Theorem 2.3 assume positive matrix dimensions, whereas the matrix in Theorem 1.5 is $g\times r$ . When $r=0$ this is a $g\times0$ matrix, so 'not very well approximable in the sense of Definition 1.3' is not defined. Waldschmidt's inequality is elementary in rank zero because its right side is $h^{\varepsilon}$ and $A(\mathbb R)_0$ is compact. The theorem should assume $r\geq1$ or dispose of $r=0$ separately.

Waldschmidt, Conjecture 1.1 ↗

Hyperplane absolute gameSuspected incorrect as written · standard repair

The game definition incorrectly guarantees a unique outcome

PDF pages 5-6 · paragraph defining HAW immediately before Theorem 2.3 · arXiv:2506.19060v1

The rules only require $\rho_{i+1}\geq\beta\rho_i$ and nesting, so the radii need not tend to zero and $\bigcap_iB_i$ need not be a singleton. The standard hyperplane-absolute game either treats failure of the radii to tend to zero as an automatic win for Alice or formulates the target condition for the intersection. The cited source's Proposition 2.3 remains valid under the standard definition, not the literal definition printed here.

Broderick-Fishman-Kleinbock-Reich-Weiss, arXiv:1106.1621 ↗

Theorem 3.1Correct after an apparently sound local repair

The rank-one exponent and uniform weak estimate are correct after a component repair

PDF pages 9-11 · Theorem 3.1, Lemma 3.2, and proof · arXiv:2506.19060v1

Replace the chosen free generator $Q$ by $2Q$ if necessary so that it lies in $E(\mathbb R)_0$ . Its logarithm ratio $\theta/\omega$ is irrational, and for a nonrational target $P$ the shift is not in $\mathbb Z(\theta/\omega)+\mathbb Z$ . Niven's nonhomogeneous Minkowski theorem gives infinitely many $q$ with distance $<|\omega|/(4|q|)$ , while $\widehat h([q]Q)=q^2\widehat h(Q)$ . This proves the uniform lower bound $\sigma_{E,K}\geq1/2$ ; the repaired Lemma 2.6 gives the matching upper bound in rank one.

Niven, Minkowski's theorem on nonhomogeneous approximation ↗

02Proofs7 reported findingsSuspected incomplete or incorrect proofs

The independent reverse proof of Theorem 2.3 is not valid as printed: its target set loses the height condition, Lemma 2.2 has an exact-relation counterexample, a crucial symbol is undefined, and the final set formula is malformed. Lemma 1.7 reverses its comparison; Lemma 2.6 omits the finite-index and torsion cosets and strengthens its hypothesis from nonrational to all points; and Theorem 3.1 assumes without justification that a free generator lies in the real identity component. Explicit repairs preserve the advertised headline results, and the cited Moshchevitin-Neckrasov theorem independently proves Theorem 2.3.

Theorem 2.3, forward implicationCorrect and complete after citing standard transference

The exponent and successive-minima calculation checks out

PDF pages 6-7 · first half of the proof of Theorem 2.3 · arXiv:2506.19060v1

With $\theta_\sigma=(n\sigma-m)/(mn(\sigma+1))$ , direct calculation gives $(1/m-\theta_\sigma)/(1/n+\theta_\sigma)=1/\sigma.$ Therefore $t=(1/n+\theta_\sigma)^{-1}\log(Q/c_2)$ yields $\|q\|<Q$ and error $c_2^{1+1/\sigma}Q^{-1/\sigma}$ ; setting $\varepsilon=n/m-1/\sigma$ gives exactly Equation (2.1). Mahler duality and comparability of covering radius with the last successive minimum justify the uniform translate estimate. The only omitted input is the standard qualitative Khintchine transference equivalence that $A$ is VWA if and only if $A^{\mathsf T}$ is VWA.

Kleinbock and Margulis, arXiv:math/9812088v2, Lemma 8.3 and Theorem 8.5 ↗

Theorem 2.3, reverse implicationSuspected incorrect as written; theorem independently verified

The HAW argument loses the bounded-denominator condition

PDF pages 7-8 · reverse proof of Theorem 2.3 · arXiv:2506.19060v1

After defining $D_{c,\mu}$ without $\|q\|<Q$ , the proof assumes from $\gamma\notin D_{c,\mu}$ that there are $q_k$ with $\|q_k\|<Q_k$ ; that inference is unavailable. The point $y_k=g_{t_k}(z_k,0)^{\mathsf T}$ then uses an undefined $z_k$ , evidently intended to be $\gamma$ . With $Q_k=c\exp((\theta_\mu+1/n)t_k)$ , the first coordinate contributes $c^{1-1/\mu}e^{\theta_\mu t_k}$ rather than the displayed $ce^{\theta_\mu t_k}$ unless the constants are renormalized. Finally, the definition of $D$ has garbled variables (' $q,m\in\mathbb Z$ ', ' $|n|<Q$ ') and the wrong existential quantifier where all bounded $q$ are required. These are not a complete proof. The cited Theorem 1(A)-(B) of arXiv:2503.21180v1 supplies the equivalence independently.

Moshchevitin and Neckrasov, Theorem 1(A)-(B), arXiv:2503.21180v1 ↗

Lemma 2.2 and rational VWA matricesSuspected incomplete as written

The exact-relation case is missing from the dynamical argument

PDF pages 5 and 7 · Lemma 2.2 and its use in the reverse proof · arXiv:2506.19060v1

The counterexample $A=0$ shows that Lemma 2.2 cannot produce local-minimum times when a lattice vector contracts forever. Since a rational matrix is VWA with infinite exponent, this omitted case lies inside the reverse implication's hypothesis. It can be handled separately: a rational orbit is finite modulo $\mathbb Z^m$ , so a shift a positive distance from that orbit immediately violates the asserted uniform approximation rate. No such case split appears in the manuscript.

Exact manuscript, arXiv:2506.19060v1 ↗

Lemma 1.7Suspected incorrect as written · an apparently sound repair

The proof reaches the opposite inequality from the one printed

PDF page 4 · proof of isogeny invariance · arXiv:2506.19060v1

Surjectivity of an isogeny on real identity components lets one lift each nonrational $P_2$ to a nonrational $P_1$ . The height comparison and Lipschitz bound send every $\kappa$ -approximation of $P_1$ to one of $P_2$ , proving $\sigma_{A_2,K}\geq\sigma_{A_1,K}$ . The manuscript writes $\leq$ . Reversing that symbol and applying the dual isogeny proves equality. The canonical-height pullback can be written directly as $\widehat h_D(\phi Q)=\widehat h_{\phi^*D}(Q)$ ; this also avoids the indirect and dimension-restricted Silverman citation.

Exact manuscript, arXiv:2506.19060v1 ↗

Theorem 1.5 and Lemma 2.5Suspected incomplete as written · standard repair

Finite-index and choice-invariance reductions are omitted

PDF page 8 · Lemma 2.5 and proof of Theorem 1.5 · arXiv:2506.19060v1

Waldschmidt's conjecture permits all of $A(K)$ , while Lemma 2.5 uses the subgroup $\Gamma$ generated by a maximal independent subset of $A(K)\cap A(\mathbb R)_0$ . This subgroup has finite index modulo torsion. Choose $N$ with $N(A(K)\cap A(\mathbb R)_0)\subseteq\Gamma$ and lift a target through multiplication by $N$ ; canonical-height quadraticity and Lipschitz continuity transfer the uniform estimate. The proof should also note that changing period bases, logarithm lifts, or a full-rank Mordell-Weil subgroup changes $J^{-1}H$ by integer or rational finite-index transformations that preserve the VWA exponent. These omissions are repairable and do not alter Theorem 1.5 for $r>0$ .

Waldschmidt, Lemma 5.1, journal pages 40-41 ↗

Lemma 2.6Suspected incomplete as written · an apparently sound repair

The bad-shift contradiction ignores rational exceptions and finite cosets

PDF page 9 · proof of Lemma 2.6 · arXiv:2506.19060v1

From $\sigma_{A,K}>r/(2g)$ the definition gives the improved exponent only for $P\notin A(K)$ , not 'for every $P\in A(\mathbb R)_0$ ' as written. Also, a maximal independent set need not generate every rational point: torsion and finitely many free cosets remain. Let $N$ kill this finite quotient. Choose $y$ in the full-dimensional Bugeaud-Harrap-Kristensen-Velani set $\operatorname{Bad}_B$ and outside the countable set of rational lifts, and take a target with $J^{-1}\xi=y/N$ . For any rational $Q$ , multiplying its lift by $N$ turns its coset and torsion term into a period, so the lower bound for $Bk+p+y$ gives $\operatorname{dist}(P,Q)\gg\widehat h(Q)^{-r/(2g)}$ . This contradicts every exponent $\mu/2>r/(2g)$ and completes the lemma, including all rational cosets.

Bugeaud-Harrap-Kristensen-Velani, Theorem 1, arXiv:0807.3863 ↗

Theorem 3.1Suspected incomplete as written · an apparently sound repair

The chosen generator may not have a real logarithm

PDF pages 9-11 · proof of Theorem 3.1 · arXiv:2506.19060v1

A generator of the free summand of $E(K)$ need not lie in $E(\mathbb R)_0$ , so the asserted $\theta\in\mathbb R$ with $\exp_E(\theta)=Q$ need not exist. Since $E(\mathbb R)/E(\mathbb R)_0$ has order at most two, replacing $Q$ by $2Q$ gives an infinite-order rational point in the identity component and only changes the uniform constant. The later sentence calling the exponential map an isometry is also too strong for an arbitrary chosen Riemannian metric; the earlier bounded-distortion statement and the displayed $\asymp$ are the correct argument. With those two replacements, the Minkowski-height proof closes.

Niven 1961, exact Proceedings article ↗

03Novelty0 reported findingsNo novelty concerns found

No non-novelty findings.

04Sources8 reported findingsSuspected incomplete or incorrect source use

The central primary sources were located and their imported conclusions checked. Waldschmidt's Conjecture 1.1 and Lemma 5.1, Moshchevitin-Neckrasov Theorem 1(A)-(B), Bugeaud-Harrap-Kristensen-Velani Theorem 1, Kleinbock-Margulis Lemma 8.3 and Theorem 8.5, Broderick et al. Proposition 2.3, and Niven's nonhomogeneous theorem all support the roles assigned to them after the repairs in this audit. Source use is nevertheless incomplete: Silverman's elliptic-curve text is cited for general abelian-variety canonical heights and does not directly give the displayed pullback comparison, while the focal paper also mislabels the Kleinbock-Margulis lemma as a theorem.

Review formExact source verified

The exact current arXiv source is version 1

Official 11-page PDF and source record · submitted June 23, 2025 · arXiv:2506.19060v1

The arXiv API and official PDF both identify version 1, dated June 23, 2025, with no later revision. All manuscript page references in this audit use that exact 11-page version.

Official arXiv:2506.19060v1 ↗

Waldschmidt [16]Applicable and sufficient

The conjecture and coefficient-height equivalence match the source

Focal PDF pages 3 and 8 · source journal page 29, Conjecture 1.1, and pages 40-41, Lemma 5.1 · arXiv:2506.19060v1

Waldschmidt's Conjecture 1.1 is the focal Conjecture 1.4 with the same dimension, rank, height, and exponent. Lemma 5.1 states the equivalence between coefficient-bounded approximation at rate $T^{-\theta}$ and canonical-height-bounded approximation at rate $h^{-\theta/2}$ for a subgroup generated by independent rational points. This is exactly the conversion used in Lemma 2.5, up to renaming the independent scale variables and the omitted finite-index reduction.

Waldschmidt, Density measure of rational points on Abelian varieties ↗

Moshchevitin-Neckrasov [12]Applicable and sufficient

The cited transference theorem independently establishes Theorem 2.3

Focal PDF pages 5 and 8 · Remark 2.4 · source Theorem 1(A)-(B), arXiv:2503.21180v1 · arXiv:2506.19060v1

In the source's notation, take the focal $m\times n$ matrix as $\Theta$ and $f(T)=T^{-(1+\delta)}$ . The conjugate function is $g(T)=T^{-1/(1+\delta)}$ . Source Theorem 1(A) makes every shift uniformly $g$ -Dirichlet when $\Theta^{\mathsf T}$ is not $f$ -approximable; after converting $T=Q^n$ , the error exponent is $n/(m(1+\delta))=n/m-\varepsilon$ . Part (B) makes the good-shift set null when the transpose is sufficiently $f$ -approximable. Varying $\delta$ is exactly the non-VWA/VWA dichotomy. Thus the source rescues the theorem, though not the focal paper's claimed independent HAW proof.

Moshchevitin and Neckrasov, arXiv:2503.21180v1 ↗

Bugeaud-Harrap-Kristensen-Velani [3]Applicable after the finite-index repair

The full-dimensional bad-shift theorem is quoted correctly

Focal PDF page 9 · Lemma 2.6 · source Theorem 1, Mathematika 56 (2010), 193-202 · arXiv:2506.19060v1

The source defines $\operatorname{Bad}_A$ by the uniform lower bound $\|Aq-x\|\geq c(x)\|q\|^{-m/n}$ for every nonzero integer $q$ and proves $\dim\operatorname{Bad}_A=n$ for every real $n\times m$ matrix. With focal dimensions $g\times r$ , this is precisely the exponent $r/g$ needed in Lemma 2.6. The source controls integer coefficient vectors; the focal proof still has to perform the omitted finite-index and torsion reduction before applying it to every rational point.

Bugeaud-Harrap-Kristensen-Velani, exact arXiv:0807.3863v2 ↗

Kleinbock-Margulis [6]Result appears correct · suspected stale locator

The dynamical correspondence is applicable but one locator is wrong

Focal PDF pages 5-7 · Lemma 2.2 and Theorem 2.3 · source Lemma 8.3 and Theorem 8.5, Inventiones 138 (1999) · arXiv:2506.19060v1

The function correspondence with Equations (8.2a)-(8.2b) is Lemma 8.3, not 'Theorem 8.3' as cited twice in the focal paper. Theorem 8.5 is the required lattice/approximation correspondence. The 2018 erratum to Kleinbock-Margulis concerns a smoothing proposition elsewhere and does not alter Lemma 8.3 or Theorem 8.5. These results support the focal forward calculation but do not imply the false exact-relation version of focal Lemma 2.2.

Kleinbock and Margulis, arXiv:math/9812088v2 ↗

Broderick-Fishman-Kleinbock-Reich-Weiss [2]Applicable with a definition correction

The HAW closure properties are correct under the source definition

Focal PDF pages 5-6 · HAW definition and Proposition 2.3 citation · source Proposition 2.3 · arXiv:2506.19060v1

Source Proposition 2.3 supplies full Hausdorff dimension and countable-intersection stability. The focal text's unique-point formula omits the standard treatment of plays whose radii do not shrink to zero, so the citation does not validate that literal formulation. Restoring the source game's convention makes the two imported properties applicable.

Broderick et al., exact arXiv:1106.1621v1 ↗

Niven [13]Applicable and sufficient

The nonhomogeneous $1/4$ estimate matches the rank-one use

Focal PDF pages 10-11 · proof of Theorem 3.1 · source Proceedings AMS 12 (1961), 992-993 · arXiv:2506.19060v1

Niven's theorem gives infinitely many integer pairs with $|q|\,|q\alpha+p-\beta|<1/4$ when $\alpha$ is irrational and $\beta\notin\mathbb Z\alpha+\mathbb Z$ . Lemma 3.2 verifies exactly those hypotheses for the logarithms of the generator and target. Together with quadraticity of the canonical height, it yields the claimed $\widehat h^{-1/2}$ rate.

Niven, exact Proceedings AMS record ↗

Silverman [15]Suspected incorrect or insufficient source use

The cited elliptic-curve text does not directly support the general abelian formulas

Focal PDF pages 3-4 and 9 · canonical-height definition, Lemma 1.7, and Section 3 setup · Silverman Chapters VIII.5 and VIII.9 · arXiv:2506.19060v1

Silverman's book treats elliptic curves, whereas pages 3-4 of the focal paper invoke it for canonical heights and isogenies of arbitrary abelian varieties. Theorem VIII.5.6 is a Weil-height functoriality estimate for morphisms and is not the displayed canonical-height comparison with $\widehat h_{\phi^*D}$ . The needed general fact is standard and stronger: canonical heights satisfy $\widehat h_D(\phi Q)=\widehat h_{\phi^*D}(Q)$ . The citation can be repaired by using a general abelian-variety height source, such as the already cited Lang chapter, and by separating the Riemannian metric from the choice of ample symmetric divisor.

Silverman, The Arithmetic of Elliptic Curves, official author page ↗

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Paper: arXiv:2506.19060v1
Authors listed: Lior Fishman, David Lambert, Keith Merrill, David Simmons
Audit date: September 26, 2026

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

Paper metadata

Abstract

Dependence graphs

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

Audit summary

Detailed mathematical audit

The Waldschmidt equivalence is correct in positive rank

The local-minimum statement fails when there is an exact relation

The displayed target set is not hyperplane absolute winning as defined

The comparison inequality is reversed although isogeny invariance survives

The matrix characterization has an unstated positive-rank hypothesis

The game definition incorrectly guarantees a unique outcome

The rank-one exponent and uniform weak estimate are correct after a component repair

The exponent and successive-minima calculation checks out

The HAW argument loses the bounded-denominator condition

The exact-relation case is missing from the dynamical argument

The proof reaches the opposite inequality from the one printed

Finite-index and choice-invariance reductions are omitted

The bad-shift contradiction ignores rational exceptions and finite cosets

The chosen generator may not have a real logarithm

The exact current arXiv source is version 1

The conjecture and coefficient-height equivalence match the source

The cited transference theorem independently establishes Theorem 2.3

The full-dimensional bad-shift theorem is quoted correctly

The dynamical correspondence is applicable but one locator is wrong

The HAW closure properties are correct under the source definition

The nonhomogeneous 1/41/41/4 estimate matches the rank-one use

The cited elliptic-curve text does not directly support the general abelian formulas

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The nonhomogeneous $1/4$ estimate matches the rank-one use