Published paper

Effective multiple equidistribution of translated measures

Michael Björklund, Alexander Gorodnik

Published version ↗Published PDF ↗Open DOI ↗arXiv record →

Publication metadata

Journal citation: International Mathematics Research Notices 2023(1), 210–242 (2023)
Publication date: 2023
DOI: 10.1093/imrn/rnab274 ↗
arXiv counterpart: arXiv:2105.05468v1

Abstract

The applicable general arbitrary-semisimple/parabolic-unipotent result is Theorem 1.3 on printed pages 213–214; Theorem 1.1 is only the special space-of-lattices statement.

Role in dependence graphs

Proof-critical source

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

This paper is included only for the following marked statement:

Theorem 1.3 · VOR printed pp. 213–214 / local PDF pp. 5–6; deduction from Theorem 2.1 on printed p. 215Effective multiple equidistribution for Wiener measures on compact abelian unipotent orbits in the general semisimple/parabolic setting.

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Version of record · International Mathematics Research Notices 2023 (2023), no. 1, 210–242

Michael Björklund, Alexander Gorodnik. Effective multiple equidistribution of translated measures. International Mathematics Research Notices 2023 (2023), no. 1, 210–242.

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Generated August 23, 2026

01Statements3 reported findingsCorrect

The abstract multiple-equidistribution theorem and its homogeneous-space specializations are correct. Theorem 1.1 is specifically the $\operatorname{SL}_{m+n}(\mathbb R)/\operatorname{SL}_{m+n}(\mathbb Z)$ compact-orbit case; Theorem 1.3, not Theorem 1.1, is the paper’s general semisimple/parabolic result.

Theorem 1.1Correct

The first main theorem is the special matrix-lattice case

Journal page 213 · Theorem 1.1 · version of record

The theorem concerns a Wiener probability measure on a compact orbit of $U_{m,n}$ in $\operatorname{SL}_{m+n}(\mathbb R)/\operatorname{SL}_{m+n}(\mathbb Z)$ . It bounds the $r$ -fold translated correlation by Sobolev norms times exponential decay in the minimum of the distances to the expanding-cone boundary and the pairwise parameter distances. Those are exactly the hypotheses and conclusion obtained from the later abstract theorem in this special setting.

IMRN version of record ↗

Theorem 1.3Correct

The general semisimple/parabolic result is Theorem 1.3

Journal pages 213–215 · Theorem 1.3 and deduction from Theorem 2.1 · version of record

Here $G$ is connected semisimple without compact factors, $P$ is a parabolic subgroup whose projection to each simple factor is proper, $U$ is its abelian unipotent radical, and the Wiener measure is supported on a compact $U$ -orbit. This is the theorem that supplies effective multiple equidistribution in the general expanding-cone setting. The version of record explicitly distinguishes it from Theorem 1.1; a citation to Theorem 1.1 for this general scope is therefore a locator error in the citing paper, not a defect in this source.

IMRN version of record ↗

Theorem 2.1Correct

The abstract correlation estimate follows from the stated equidistribution inputs

Journal pages 214–215 · assumptions (EQ1)–(EQ2) and Theorem 2.1 · version of record

For a $U$ -ergodic measure with discrete spectrum, the theorem assumes effective one-translate equidistribution, polynomial mixing along $U$ , and explicit algebra and norm controls. It concludes decay for every higher translated correlation in the minimum of the individual equidistribution scales and pairwise $T$ -separations. The proof’s induction records every Sobolev-degree and exponent loss, so the constants depend only on the announced order and norm data.

IMRN version of record ↗

02Proofs4 reported findingsCorrect

The Fourier reduction, one-factor estimate, root-direction selection, and recursive higher-correlation bound are correct and complete. Two uniquely determined variable-name errors are harmless.

Sections 3–4Correct and complete

Fourier expansion and the base estimate correctly start the induction

Journal pages 215–223 · Sections 3–4.3 · version of record

The discrete $U$ -spectrum expands the Wiener density into eigencharacters with absolutely summable coefficients. For a nontrivial eigencharacter, averaging along a one-parameter subgroup converts its oscillation into a term controlled jointly by (EQ1) and (EQ2); optimizing the averaging length gives a positive decay exponent. A root direction maximizing the relative adjoint expansion of the translate parameters is then selected, and its ordered expansion sizes provide the scale separation required for the inductive step.

IMRN version of record ↗

Proposition 4.4 and Lemma 4.5Correct and complete

The recursive error decomposition closes for every correlation order

Journal pages 225–232 · Proposition 4.4, Lemma 4.5, and proof of Theorem 2.1 · version of record

Proposition 4.4 decomposes the averaged correlation at a split index into an averaging error, a mixing error, and products of lower-order errors, with the announced Sobolev losses. Lemma 4.5 uses the ordered adjoint-expansion sizes to choose a split whose adjacent gap is large relative to the overall separation. Choosing the averaging length as a power of that gap makes every term a negative power of the controlling scale; induction on $r$ then proves Theorem 2.1.

IMRN version of record ↗

Lemma 4.1 proofTypo

An undefined test-function symbol should be $\varphi$

Journal page 220 · proof of Lemma 4.1 · version of record

The third expression in the displayed computation ends with the undefined factor $\psi$ , in $\nu(\psi_\xi\circ\exp(-w)\cdot\psi)$ . The unique correction is to replace that last factor by the test function $\varphi$ . Then $U$ -invariance and the eigenfunction identity give $\exp(w)_*\nu_\xi=\xi(-w)\nu_\xi$ exactly as the following line states.

IMRN version of record ↗

Introductory description of the decay parameterTypo

The final translate parameter is misindexed

Journal page 212 · paragraph immediately before Theorem 1.1 · version of record

The paragraph says that the improved estimate depends on $t_1,\ldots,t_1$ and the pairwise distances. The unique correction is $t_1,\ldots,t_r$ , as shown in the definition of the decay parameter immediately below and in every theorem and proof use. No mathematical assertion changes.

Effective multiple equidistribution of translated measures

Publication metadata

Abstract

Role in dependence graphs

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

Audit summary

Version of record · International Mathematics Research Notices 2023 (2023), no. 1, 210–242

The first main theorem is the special matrix-lattice case

The general semisimple/parabolic result is Theorem 1.3

The abstract correlation estimate follows from the stated equidistribution inputs

Fourier expansion and the base estimate correctly start the induction

The recursive error decomposition closes for every correlation order

An undefined test-function symbol should be φ\varphiφ

The final translate parameter is misindexed

An undefined test-function symbol should be $\varphi$