arXiv:2504.02258v1

Submanifold-genericity of $\mathbb{R}^d$ -actions and uniform multiplicative Diophantine approximation

Prasuna Bandi, Reynold Fregoli, Dmitry Kleinbock

math.NTmath.DS37A2537A4411J1311K60

Paper metadata

Original submission: April 3, 2025
Claimed MSC: 37A25, 37A44, 11J13, 11K60
arXiv categories: math.NT, math.DS
Paper license: arXiv non-exclusive distribution license ↗

Abstract

In this paper, we prove a new ergodic theorem for $\mathbb{R}^d$ -actions involving averages over dilated submanifolds, thereby generalizing the theory of spherical averages. Our main result is a quantitative estimate for the error term of such averages valid for smooth functions under some effective mixing assumptions on the action. With the aid of this theorem, we investigate multiplicative-type Dirichlet-improvability for $(m\times n)$ -matrices with real coefficients. In particular, we establish that almost all matrices are uniformly approximable by the function $x\mapsto x^{-1}(\log x)^{-1+\varepsilon}$ for any $\varepsilon>0$ . Results of this type motivate a question which can be thought as a strengthening of Littlewood's conjecture in multiplicative Diophantine approximation.

Dependence graphs

Proof lineage

Submanifold-genericity of $\mathbb{R}^d$-actions and uniform multiplicative Diophantine approximation

Statement-restricted proof-dependence graph for the submanifold genericity theorems, the multiplicative Dani correspondence, the full-measure half of the Khintchine-type theorem, and Proposition 1.7. It includes the uncited cusp-volume input required on p. 28, its independent Schmidt 1957 proof repair, and two source-use failures: BEG20 is applied after dropping its strong-spectral-gap hypothesis, and BG23 is cited with the wrong theorem number.

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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 23, 2026

01Statements4 reported findingsContains wrong statements

Theorem 1.3 is false for the stated class of semisimple groups and arbitrary lattices: reducible product lattices give invariant factor functions along admissible submanifolds. Requiring the full action to have strong spectral gap, for example by taking an irreducible lattice under the standard hypotheses, repairs it. Theorem 1.4 and the multiplicative conclusions are correct after the verified internal and source repairs recorded below. The displayed measure in Theorem 1.8 needs a local normalization correction.

Theorem 1.3Incorrect as written · verified hypothesis repair

Reducible product lattices contradict submanifold genericity

PDF page 3 · Theorem 1.3 and estimate (1.4) · arXiv:2504.02258v1

Take $G=G_1\times G_2$ with $G_i=\operatorname{SL}_3(\mathbb R)$ and choose cocompact lattices $\Gamma_i<G_i$ , setting $\Gamma=\Gamma_1\times\Gamma_2$ . Every simple factor has real rank $2$ , so the printed hypotheses hold, and $X=X_1\times X_2$ is compact. Choose nonzero $H\in\mathfrak a_1$ and let $M=\{s(H,0):1\leq s\leq2\}$ , an allowed compact one-dimensional submanifold. For any nonconstant $v\in C^\infty(X_2)$ , the function $\varphi(x_1,x_2)=v(x_2)$ belongs to $C_c^\infty(X)$ and every element of $tM$ fixes it. Thus the average in (1.4) equals $v(x_2)$ for every $t$ , while its claimed limit is $\mu_2(v)$ ; these differ on a positive-measure set. The effective estimate fails as well. The cited BEG theorem explicitly requires strong spectral gap, which this reducible product action lacks. Adding that hypothesis, or imposing the usual irreducibility assumptions that supply it, is the verified repair.

Björklund–Einsiedler–Gorodnik, Theorem 1.1 ↗

Theorem 1.4Correct

Genericity of compact $U_{m,n}$ -orbit measure

PDF pages 3 and 9–16 · Theorem 1.4 and Sections 2–3 · arXiv:2504.02258v1

Here $G=\operatorname{SL}_{m+n}(\mathbb R)$ is simple and $m+n\geq3$ , so the spectral and irreducibility obstruction affecting Theorem 1.3 is absent. The exact general BG23 theorem applies to the Haar probability on a compact $U_{m,n}$ -orbit, whose density has Wiener norm $1$ . Compactness of $M\subset E$ makes its distance from the cone walls uniformly comparable to the pairwise-separation parameter. The corrected even-moment argument then gives the claimed genericity and, by taking arbitrarily large even moments, the rate $t^{-k/2+\delta}$ .

Björklund–Gorodnik, general Theorem 1.3 ↗

Proposition 1.7 and Theorems 1.8–1.9Correct

Uniform multiplicative approximation conclusions survive the proof repairs

PDF pages 6 and 17–30 · Proposition 1.7, Theorems 1.8–1.9, and Sections 4–6 · arXiv:2504.02258v1

The multiplicative Dani correspondence and the internally proved transference lemma correctly reduce Theorem 1.8 to Theorem 1.4. For Theorem 1.9, the full-measure branch follows after replacing the polytope with a smooth compact patch and supplying the omitted fixed-ratio cusp-annulus estimate; the moment exponents then give the threshold $\lambda<m+n-2$ . In the zero-measure branch, the corrected product-region volume and lattice-point count give $O((\log T)^{m+n-2-\lambda})$ . Proposition 1.7 follows from Minkowski after shrinking the two radii within the strict slack $c>m!n!/(m^mn^n)$ . These verified repairs preserve the exact parameter ranges of all three conclusions.

Exact arXiv version 1 ↗

Theorem 1.8Minor formal correction

The Lebesgue measure must be normalized on a fundamental cube

PDF page 6 · Theorem 1.8 · arXiv:2504.02258v1

The set is written as a subset of $\mathbb R^{m\times n}$ , whose Lebesgue measure is infinite when the set is conull, so the displayed equality $\operatorname{Leb}(D_{m,n}^{\times}(c\psi_1))=1$ is not literal. Replace it by $\operatorname{Leb}(D_{m,n}^{\times}(c\psi_1)\cap[0,1)^{mn})=1,$ or state that the complement has Lebesgue measure zero. Integer-translation invariance and the proof on the compact $U_{m,n}$ -orbit make these formulations equivalent. The correction is local and changes no application.

02Proofs8 reported findingsContains incorrect or incomplete proofs

The proof of Theorem 1.3 applies BEG outside its strong-spectral-gap hypothesis and the theorem is genuinely false in that scope. The later proofs have repairable but substantive defects: the chosen averaging patch has corners, the required cusp-annulus measure estimate is absent, two mollifier calculations are false as printed, the zero-measure volume formula is wrong, and the Minkowski argument proves non-strict inequalities. Exact repairs establish the unaffected conclusions. Two literal notation errors are reported separately.

Proof of Theorem 1.3Incorrect as written · no repair at the printed scope

The cited multiple-mixing theorem lacks a required hypothesis

PDF pages 9–10 · Theorem 2.3 and proof of Theorem 1.3 · arXiv:2504.02258v1

BEG Theorem 1.1 assumes that the acting semisimple group has strong spectral gap on $L_0^2(G/\Gamma)$ . The paper replaces that assumption by ‘every simple factor has rank at least $2$ ’. Property $(T)$ supplies a gap only when the corresponding factor action is ergodic; for a reducible product lattice, one factor fixes all functions of the other factor. Thus estimate (2.5) is unavailable and the counterexample in the Statements dimension shows that the lost hypothesis cannot be reconstructed. Repair classification: no repair at the printed scope; adding strong spectral gap or an appropriate irreducibility hypothesis repairs the proof and statement.

BEG exact arXiv version 2, Theorem 1.1 ↗

Full-measure proofs of Theorems 1.8–1.9Incorrect as written · verified repair

The averaging polytope is not an allowed $C^1$ submanifold

PDF pages 22 and 26–28 · definition of $M_\sigma$ and Lemma 5.4 · arXiv:2504.02258v1

$M_\sigma$ is cut out by several simultaneous coordinate inequalities and has corners, whereas Theorems 1.4 and 2.2 assume a $C^1$ submanifold, possibly with ordinary boundary. The same issue appears in Lemma 5.4, whose parameter set $J$ is a closed polytope although Lemma 3.1 assumes a bounded open chart domain. Replace $M_\sigma$ by the closure of a small Euclidean ball in the affine slice $E_1$ , compactly contained in $E$ , and take the interior ball as $J$ . The parametrization remains affine, the dimension is $m+n-2$ , and every mixing, collision-volume, and cusp-intersection estimate used later is unchanged up to fixed constants. Repair classification: Verified repair.

Proof of Theorem 1.9Incomplete as written · verified external-input repair

The cusp-annulus measure estimate is missing

PDF page 28 · paragraph beginning ‘For $\Omega_t$ as in (5.4)’ · arXiv:2504.02258v1

The proof asserts $\mu(\Omega_t)\asymp e^{-(m+n)R(t)}$ and says this follows from Lemma 5.3. That lemma only bounds the support of a mollified indicator; it contains no Haar-measure estimate. Put $D=m+n\geq3$ and $r=e^{-R(t)}$ . The primitive Siegel mean formula together with Rogers' second-moment estimate gives $\mu\{\Lambda:\delta(\Lambda)<r\}=\frac{2^{D-1}}{\zeta(D)}r^D+O_D(r^{2D}).$ Subtracting the same formula at $r/e$ yields $\mu\{r/e\leq\delta<r\}\asymp_Dr^D$ , exactly the missing claim. This repairs the full-measure branch, but it is a substantive proof-critical input and cannot be attributed to Lemma 5.3.

C. A. Rogers, Mean Values over the Space of Lattices, Theorems 4–5 (printed pages 251–253) ↗

Lemmas 5.1–5.3Incorrect as written · verified repair

The mollifier derivative and support calculations are false as printed

PDF pages 23–26 · equations (5.1)–(5.5) and Lemmas 5.1–5.3 · arXiv:2504.02258v1

From $\rho_r(\psi(x))=r^{-D_G}\rho(r^{-1}x)/f(x)$ , differentiating along $\gamma_\alpha(x,s)=\psi^{-1}(\exp(s\alpha)\psi(x))$ requires the product and quotient rules. The displayed proof instead differentiates $\rho(r^{-1}\gamma_\alpha/f(\gamma_\alpha))$ , moving the density into the argument of $\rho$ and omitting its derivative. Also, coordinate support $x\in B_r(0)$ only gives $\operatorname{supp}\rho_r\subset B_{C_1r}(\mathrm{id})$ , not the opening claim $B_r(\mathrm{id})$ , and Lemma 5.3 later reuses the smaller ball. In that lemma the two uniquely determined notation corrections are $\varphi_{r,t}\to\varphi_{t,r}$ and, in the quantified lower bound, $w\to w'$ . Applying the correct chain and quotient rules gives $\|D^j\rho_r\|_\infty=O(r^{-D_G-j})$ because $f$ and $1/f$ have bounded derivatives on the chart. Consistently use $B_{C_1r}$ and initially require $C_1r<r_0$ ; all convolution and Sobolev estimates then follow with changed fixed constants. Repair classification: Verified repair.

Zero-measure proof of Theorem 1.9Incorrect as written · verified repair

The displayed product-region volume is not the stated equality

PDF pages 29–30 · final counting argument · arXiv:2504.02258v1

For $0<\varepsilon\leq2^{-m}$ , the exact volume of $\{x\in[-1/2,1/2]^m:\prod_i|x_i|<\varepsilon\}$ is $2^m\varepsilon\sum_{j=0}^{m-1}\frac{(\log(2^{-m}\varepsilon^{-1}))^j}{j!},$ not $2^m\varepsilon[(\log(2^{-m}\varepsilon^{-1}))^{m-1}+1]$ except when $m=2$ . The needed upper bound $O_m(\varepsilon(1+|\log\varepsilon|)^{m-1})$ nevertheless follows. In the preceding union, replace the unsupported bound $|p|<|q|$ by $|p|\leq n|q|+1$ in the supremum norm after choosing nearest integers; this changes only a fixed counting constant. The corrected count is still $O((\log T)^{m+n-2-\lambda})$ . Repair classification: Verified repair.

Proof of Proposition 1.7Incorrect as written · verified repair

Minkowski yields weak inequalities while the target set is strict

PDF page 30 · equations (6.1)–(6.2) and conclusion · arXiv:2504.02258v1

The closed convex body gives $\prod_i|Y_iq-p_i|\leq c/T$ and, when every $q_j\neq0$ , $\Pi_+(q)\leq T$ , but $S_{m,n}^{\times}(c\psi_1,T)$ is defined using both strict inequalities. Choose $c_0$ and $\eta$ with $\frac{m!n!}{m^mn^n}<c_0\eta<c_0<c,$ use the first radius with $c_0$ and replace the second radius by $n(\eta T)^{1/n}$ . The body still has volume greater than $2^{m+n}$ , while the resulting products are $<c/T$ and $<T$ for all sufficiently large $T$ ; the cases with zero coordinates are already easier. Repair classification: Verified repair.

Proof of Theorem 2.2Typo

The mean-value point has the wrong sign

PDF page 11 · proof of Theorem 2.2 · arXiv:2504.02258v1

The printed point $\theta_j s-(1-\theta_j)s'$ should be $\theta_j s+(1-\theta_j)s'$ . The latter lies on the segment joining $s'$ to $s$ , as required by the mean value theorem. Every downstream estimate uses only membership in that convex segment, so the correction is unique and harmless.

Proof of Theorem 1.9Typo

The subscripts of the approximation function are reversed

PDF pages 29–30 · three occurrences in the zero-measure argument · arXiv:2504.02258v1

Replace each $\psi_{\lambda,1}$ by $\psi_{1,\lambda}$ . Only $\psi_{1,\lambda}(x)=x^{-1}(\log x)^{-\lambda}$ is defined, and that substitution is exactly what the displayed estimate uses.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

04Sources5 reported findingsContains incorrect or incomplete source use

The BEG theorem is invoked without its strong-spectral-gap hypothesis, producing a false headline theorem. The BG23, BG20, and FK inputs are mathematically sufficient after correcting the BG23 theorem number and the FK proposition/corollary label, while the fixed-ratio cusp-annulus estimate is a necessary uncited input. Thus the focal source chain is neither correct nor complete as printed.

BEG Theorem 1.1Incorrect source use

Strong spectral gap is indispensable, not implied by the printed rank condition

Target PDF pages 3 and 9–10 · Theorems 1.3 and 2.3 · arXiv:2504.02258v1

The exact BEG theorem assumes that the $G$ -action on $G/\Gamma$ has strong spectral gap, meaning that its restriction to every noncompact simple factor is isolated from the trivial representation on $L_0^2$ . The target drops this and assumes only rank at least $2$ for each factor. For a product lattice, each factor fixes the nonconstant functions pulled back from the other quotient, so the source hypothesis fails. The explicit counterexample in the Statements dimension proves that the source conclusion cannot be extended as claimed. Adding strong spectral gap or an appropriate irreducibility condition is necessary.

Quantitative Multiple Mixing, arXiv:1701.00945v2, Theorem 1.1 ↗

BG23 general Theorem 1.3Applicable after verified citation repair

The cited theorem number is too narrow, but the same paper has the needed general result

Target PDF page 10 · Theorem 2.4 · arXiv:2504.02258v1; BG23 VOR pages 213–214

The target attributes its arbitrary-lattice statement to BG23 Theorem 1.1. In the exact version of record, Theorem 1.1 is the special $\operatorname{SL}_{m+n}(\mathbb R)/\operatorname{SL}_{m+n}(\mathbb Z)$ case. The required parabolic-unipotent statement is general Theorem 1.3. It applies here because $G=\operatorname{SL}_{m+n}(\mathbb R)$ is simple, every lattice is irreducible in the relevant sense, $U_{m,n}$ is the abelian unipotent radical, and Haar probability on its compact orbit is a Wiener measure of norm $1$ . Replacing the locator ‘Theorem 1.1’ by ‘Theorem 1.3’ verifies the source use.

Effective Multiple Equidistribution of Translated Measures, version of record ↗

Fixed-ratio cusp-annulus asymptoticMissing proof-critical source

A proof-critical geometry-of-numbers input is absent

Target PDF page 28 · proof of Theorem 1.9 · arXiv:2504.02258v1

Neither Lemma 5.3 nor any represented citation proves the asserted estimate $\mu(\Omega_t)\asymp e^{-(m+n)R(t)}$ . This is not a support property of the mollifier; it is a Haar-measure theorem for short primitive lattice vectors. Rogers' primitive first- and second-moment formulas, Theorems 4–5, supply the exact asymptotic and repair the argument, as detailed in the Proofs dimension. Because this external result is necessary for the lower bound in Lemma 5.4 and hence the full-measure half of Theorem 1.9, it must be represented as a direct dependency rather than treated as routine algebra.

C. A. Rogers, Mean Values over the Space of Lattices, Theorems 4–5 (printed pages 251–253) ↗

BG20 Proposition 6.2 and FK24 Lemma 4.1 / Proposition 4.4 / Corollary 4.5Applicable and sufficient after verified citation repair

The covering and multiplicative-correspondence inputs are applicable after one locator correction

Target PDF pages 12 and 17–18 · Lemma 3.4, Lemma 4.1, and Proposition 4.2 · arXiv:2504.02258v1

BG20 Proposition 6.2 is the exact finite cluster decomposition used in Lemma 3.4; its proof uses only the metric triangle inequality, so the target's extension from the source's group notation to a metric space is valid. The exact source available when the target appeared is arXiv:2211.04523v3: its Lemma 4.1 gives the $\psi$ – $R$ parametrization, Proposition 4.4 gives precisely the multiplicative Dani correspondence used in Proposition 4.2 and Corollary 4.3, and Corollary 4.5 is the cited consequence. The target's locator ‘Proposition 4.5’ must therefore be corrected to ‘Corollary 4.5’. Its sign convention for the last $n$ diagonal exponents is a faithful renaming. Version 4 appeared later on April 3, 2025 and is outside the represented source boundary; the unrelated defects found elsewhere in version 3 do not affect Lemma 4.1, Proposition 4.4, or Corollary 4.5 as used here.

Fregoli–Kleinbock, exact cited arXiv version 3 ↗

Minkowski convex body theoremApplicable and sufficient after the verified internal repair

The theorem applies; strictness is an internal slack issue

Target PDF page 30 · proof of Proposition 1.7 · arXiv:2504.02258v1

The body $\Xi(Y,T,c)$ is convex, centrally symmetric, and has the displayed volume $2^{m+n}m^mn^nc/(m!n!)$ . Minkowski therefore supplies a nonzero integer point once $c>m!n!/(m^mn^n)$ . The source theorem itself is sufficient. The mismatch between weak inequalities from the closed body and strict inequalities in $S_{m,n}^{\times}$ is repaired entirely inside the focal proof by the two slack parameters described in the Proofs dimension.

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Paper: arXiv:2504.02258v1
Authors listed: Prasuna Bandi, Reynold Fregoli, Dmitry Kleinbock
Audit date: August 23, 2026

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Submanifold-genericity of Rd\mathbb{R}^dRd-actions and uniform multiplicative Diophantine approximation

Paper metadata

Abstract

Dependence graphs

Submanifold-genericity of $\mathbb{R}^d$-actions and uniform multiplicative Diophantine approximation

Audit summary

Detailed mathematical audit

Reducible product lattices contradict submanifold genericity

Genericity of compact Um,nU_{m,n}Um,n​-orbit measure

Uniform multiplicative approximation conclusions survive the proof repairs

The Lebesgue measure must be normalized on a fundamental cube

The cited multiple-mixing theorem lacks a required hypothesis

The averaging polytope is not an allowed C1C^1C1 submanifold

The cusp-annulus measure estimate is missing

The mollifier derivative and support calculations are false as printed

The displayed product-region volume is not the stated equality

Minkowski yields weak inequalities while the target set is strict

The mean-value point has the wrong sign

The subscripts of the approximation function are reversed

Strong spectral gap is indispensable, not implied by the printed rank condition

The cited theorem number is too narrow, but the same paper has the needed general result

A proof-critical geometry-of-numbers input is absent

The covering and multiplicative-correspondence inputs are applicable after one locator correction

The theorem applies; strictness is an internal slack issue

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Submanifold-genericity of $\mathbb{R}^d$ -actions and uniform multiplicative Diophantine approximation

Genericity of compact $U_{m,n}$ -orbit measure

The averaging polytope is not an allowed $C^1$ submanifold