arXiv:2608.30735v2

Density of the multidimensional Lagrange spectrum

math.NTmath.DS11J0611J1337A1737A25

Paper metadata

Original submission: August 31, 2026
Latest revision: September 1, 2026 · version 2
Claimed MSC: 11J06, 11J13, 37A17, 37A25
arXiv categories: math.NT, math.DS
Comments: 14 pages; the proof of Theorem 1.2 and a few misprints have been corrected

Abstract

The Lagrange spectrum is a classical object in number theory, defined as the set of values of $\liminf_{q\to\infty} q\, \mathrm{dist}(qα,\mathbb{Z})$ where $α$ runs through irrational numbers. It has a complicated structure, with the discrete part, Hall's ray, and a transitional part in between. One can similarly define Lagrange spectrum in the multidimensional set-up, and until now not much has been understood about it. In this paper we prove that, unlike in the one-dimensional case, the closure of the multidimensional Lagrange spectrum is equal to the interval between $0$ and its supremum. The proof relies on a correspondence between Diophantine approximation and dynamics on the space of unimodular lattices and proceeds by studying a dynamical counterpart of the Lagrange spectrum that we call dynamical Lagrange spectrum. The latter is shown to be equal to the interval between $0$ and its maximum by means of an argument utilizing the higher rank nature of the set-up. A passage from full dynamical spectrum to the density of the Diophantine spectrum is achieved by applying equidistribution of expanding translates of horospheres in the space of lattices.

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Audited against arXiv v2

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.

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Detailed mathematical audit

Generated October 24, 2026

01Statements4 reported findingsNo statement issues found

The four main theorems follow from the quantitative lattice correspondence, the repeated-weight centralizer deformation, and a verified horospherical shadowing argument.

Theorem 1.2No issue found

The exact quantitative Dani correspondence holds for the stated norms

PDF pp. 4 and 5–6 · arXiv:2608.30735v2

For $q\ne0$ , put $a_q=\operatorname{dist}_{\rho}(Aq,\mathbb Z^m)$ and $b_q=\sigma(q)$ . The minimum of $\max(e^{t/m}a_q,e^{-t/n}b_q)$ occurs at $t_q=mn\log(b_q/a_q)/(m+n)$ and has $(m+n)$ -th power $a_q^m b_q^n$ . The covering radius bounds $a_q$ , so $t_q\to\infty$ when $q\to\infty$ and $a_q>0$ . This proves one liminf inequality. Conversely, shortest vectors along any times approaching the dynamical liminf satisfy $a_q\le e^{-t/m}(L+\varepsilon)^{1/(m+n)}$ and $b_q\le e^{t/n}(L+\varepsilon)^{1/(m+n)}$ ; multiplication gives the opposite inequality, and absence of an exact rational relation forces their denominators to infinity. If an exact nonzero relation exists, its integer multiples give Diophantine liminf zero and its contracting lattice vector gives dynamical liminf zero. If the Diophantine liminf is zero without an exact relation, the minimizing times already give dynamical liminf zero. Thus the zero case and arbitrary nonsmooth norms are covered without a differentiability assumption.

Theorem 1.3No issue found

Every intermediate dynamical value, including the maximum, is attained

PDF pp. 5 and 7–11 · arXiv:2608.30735v2

Mahler compactness applied after shifting nearly maximal trajectories gives a lattice whose entire forward orbit has shortest-vector $(m+n)$ -th power at least the spectrum supremum; continuity at each fixed time then proves attainment. For general dimension $d$ , repeated weights give a two-dimensional scalar block. On that block choose a basis whose first vector is the projection of a limiting shortest vector, and set $h_s=s\operatorname{diag}(\theta_s,I_{d-2})$ , where $\theta_s$ fixes that first vector and multiplies its complementary basis vector by $s^{-d}$ . If the projection is zero, any block basis works. Then $\det h_s=1$ , $h_s$ commutes with the flow, and $h_sv=sv$ . The operator-norm comparison makes $s\mapsto\lambda^F_{\eta}(h_sx)$ continuous, while the limiting vector gives an upper bound $s^d\lambda^F_{\eta}(x)$ . The intermediate value theorem supplies every positive target below the maximum. Mixing gives zero through unbounded generic trajectories. The zero-supremum case is immediate; the positive-supremum compactness proof is uniform over its selected sequence.

Theorems 1.4 and 4.3No issue found

The shadowing construction gives density inside every expanding horospherical orbit

PDF pp. 5 and 11–13 · arXiv:2608.30735v2

Choose a small centralizer neighborhood on which the value at $x_0$ stays in the target interval, and a product neighborhood $Q=BB^0B^-x_0$ with null boundary. Given an arbitrary open set $V$ in the expanding subgroup, first choose $h_0$ , a relatively compact open $W$ , and a small neighborhood $N$ with $NW h_0\subset V$ . Equidistribution gives $g_thh_0x=uzu^-x_0$ for arbitrarily large suitable $t$ , with $h\in W$ and $u\in B$ . Set $u'=(g_{-t}u^{-1}g_t)hh_0$ . Uniform contraction of the bounded set $B^{-1}$ puts $u'$ in $V$ , and $g_tu'x=zu^-x_0$ . The centralizer normalizes the contracting subgroup, so contracting invariance and flow invariance give $\lambda^F_{\eta}(u'x)=\lambda^F_{\eta}(zx_0)$ in the prescribed interval. This uses one finite shadowing correction followed by exact contracting invariance; no infinite-time approximation or unproved continuity transverse to the centralizer is required.

Theorem 1.1No issue found

The multidimensional Diophantine spectrum has the asserted closure and local density

PDF p. 3 and conclusion on p. 13 · arXiv:2608.30735v2

For the two-block flow, $\max(m,n)>1$ supplies repeated weights and its full expanding horospherical subgroup is exactly $U=\{u_A\}$ . Theorems 1.3 and 4.3 therefore put dynamical values arbitrarily near every point of $[0,L^F_{\eta}]$ in every nonempty open set of matrices. Theorem 1.2 identifies their values with the Diophantine constants, proving both equality of the two suprema and the claimed density. Intervals meeting the endpoints are handled by approaching positive interior values. This does not assert that the Diophantine supremum itself is attained. The discussion of weighted approximation on pp. 13–14 correctly distinguishes its generally smaller matrix subgroup from the full expanding horospherical subgroup, so the final argument is not extended to unequal weights without justification.

02Proofs4 reported findingsNo proof issues found

All substantive proof steps and the material cited equidistribution input are verified. Two forced notation corrections do not change the arguments.

Proposition 4.2 and its use in Theorem 4.3No issue found in the reviewed proof

The primary equidistribution theorem applies with its exact hypotheses

PDF pp. 12–13 · arXiv:2608.30735v2

Kleinbock, Badly approximable systems of affine forms, Proposition 2.4, requires a semisimple flow generator, mixing on a finite-volume quotient, a bounded measurable horospherical set, a bounded target with measure-zero boundary, and a compact set of base points. The diagonal generator here is semisimple; the nontrivial diagonal flow is unbounded in the simple group and mixing by the source's Moore criterion in Section 2.2. The product neighborhoods may be chosen small and relatively compact with piecewise smooth boundary, and the base-point compact set is the singleton $\{h_0x\}$ . Taking the error below the positive product of the two measures gives a hit for every sufficiently large time, exactly what is needed for the simultaneous contraction requirement. The stronger assertion is not inferred from ergodicity alone.

Propositions 3.1–3.3, Lemma 3.5 and Proposition 3.6No issue found in the reviewed proof

The compactness, limiting-vector and continuity steps are justified

PDF pp. 7–10 · arXiv:2608.30735v2

For a positive dynamical constant, Mahler compactness bounds the whole forward trajectory after adding its finite initial segment. Shortest vectors along a liminf sequence then remain in a compact annulus, and convergent lattice bases show that their nonzero limit belongs to the limiting lattice. Continuity of the shortest-vector function identifies its length with that lattice's minimum. The centralizer estimate $\|z^{-1}\|^{-d}\lambda^F_{\eta}(x)\le\lambda^F_{\eta}(zx)\le\|z\|^d\lambda^F_{\eta}(x)$ proves the required orbitwise continuity, also when the constant is zero. The deformation only needs an upper bound at $s$ and the exact value at $1$ ; no unjustified equality of a shortest-vector branch throughout the deformation is used. Proposition 3.1's compactness argument handles positive supremum, and a zero supremum is already attained because every constant is nonnegative.

Lemma 4.1Suspected typo

The comparison must use the reference lattice at time t

PDF p. 11 · first displayed two-sided estimate in the proof · arXiv:2608.30735v2

Both outer factors are printed with $\delta_{\eta}(ux)$ . Replace those two occurrences by $\delta_{\eta}(g_tx)$ . The preceding identity is $g_tux=(g_tug_{-t})g_tx$ , and the operator-norm inequality already used in Lemma 3.5 uniquely determines this correction. Thus the correct estimate is $\|(g_tug_{-t})^{-1}\|^{-1}\delta_{\eta}(g_tx)\le\delta_{\eta}(g_tux)\le\|g_tug_{-t}\|\delta_{\eta}(g_tx)$ . Contraction makes both factors tend to one; the global shortest-vector bound then gives the displayed difference tending to zero. The lemma and every subsequent use are unchanged.

Lemma 3.4Suspected typo

The determinant calculation misnames the already defined block matrix

PDF p. 9 · determinant display below the definition of $h_s$ · arXiv:2608.30735v2

The determinant display uses the undefined symbol $h_{\theta}$ twice. Both occurrences are $h'_s$ , defined immediately above with determinant $s^{-d}$ . The calculation is $\det(h_s)=\det(sh'_s)=s^d\det(h'_s)=1$ . This is a forced variable-name correction, not a modification of the deformation.

03Novelty0 reported findingsNo novelty concerns found

No non-novelty findings.

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Paper: arXiv:2608.30735v2
Authors listed: Dmitry Kleinbock
Audit date: October 24, 2026

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Density of the multidimensional Lagrange spectrum

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The exact quantitative Dani correspondence holds for the stated norms

Every intermediate dynamical value, including the maximum, is attained

The shadowing construction gives density inside every expanding horospherical orbit

The multidimensional Diophantine spectrum has the asserted closure and local density

The primary equidistribution theorem applies with its exact hypotheses

The compactness, limiting-vector and continuity steps are justified

The comparison must use the reference lattice at time t

The determinant calculation misnames the already defined block matrix

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