Published paper

Quantitative multiple mixing

Michael Björklund, Manfred Einsiedler, Alexander Gorodnik

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

Publication metadata

Journal citation: Journal of the European Mathematical Society 22(5), 1475–1529 (2020)
Publication date: 2020
DOI: 10.4171/JEMS/949 ↗
arXiv counterpart: arXiv:1701.00945v2

Abstract

Theorem 1.1 supplies the quantitative multiple-mixing estimate invoked by the focal paper. Its strong-spectral-gap hypothesis is essential and is omitted from the focal restatement.

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 statements:

Theorem 1.1 · arXiv v2 PDF p. 3; proof §§2–3, PDF pp. 9–22Exponential mixing of every order for a semisimple action with strong spectral gap.
Sobolev norm system (2.10)–(2.14) · arXiv v2 PDF pp. 11–12Weighted Sobolev norms with embedding, action-growth, and product estimates used throughout the coupling induction.

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Exact reviewed source

arXiv:1701.00945v2 · explicit fallback for the inaccessible JEMS version of record

Michael Björklund, Manfred Einsiedler, Alexander Gorodnik. Quantitative multiple mixing. Journal of the European Mathematical Society 22 (2020), no. 5, 1475–1529. Reviewed in the final arXiv:1701.00945v2 form.

The exact JEMS version of record was not publicly downloadable during this audit. The complete final arXiv version 2 was therefore reviewed as the explicit open proof-bearing fallback; this report does not claim inspection of the publisher file.

Open audited source ↗

Generated August 23, 2026

01Statements4 reported findingsCorrect

The exponential multiple-mixing theorems for real, $S$ -algebraic, and adelic homogeneous actions, the coupling-uniform forms, and the approximate-configuration consequence are correct at their printed scope. In particular, the real homogeneous-space theorem explicitly assumes strong spectral gap; the paper never asserts that result without this hypothesis.

Theorems 1.1 and 1.2Correct

Real multiple mixing is stated with the necessary strong-spectral-gap hypothesis

PDF pages 2–4 · definition of strong spectral gap and Theorems 1.1–1.2 · arXiv:1701.00945v2

The action of the connected semisimple group $G$ on $X=\Gamma\backslash L$ is expressly required to have strong spectral gap, meaning that the representation on $L^2_0(X)$ restricts to every noncompact simple factor with the trivial representation isolated. Under precisely that hypothesis, Theorem 1.1 bounds every $k$ -point correlation by an exponentially decaying function of the minimum pairwise group distance, and Theorem 1.2 gives the same estimate uniformly over diagonal-invariant couplings. The binary matrix-coefficient estimate used to start the proof is available under that same hypothesis, so the statement and its proof have matching scope.

Final arXiv version 2 ↗

Corollary 1.3Correct

The approximate lattice-configuration consequence follows

PDF pages 4–5 and Section 3 · Corollary 1.3 and proof · arXiv:1701.00945v2

For an irreducible lattice in a semisimple group without compact factors, the required strong spectral gap is available. Smooth approximate identities on the quotient have controlled Sobolev norms; applying Theorem 1.1 to their translates makes the multiple correlation positive once the minimum pairwise distance is a sufficiently large multiple of $\log(1/\varepsilon)$ . Unfolding that positive integral produces the asserted lattice elements and common translate with error below $\varepsilon$ .

Final arXiv version 2 ↗

Theorems 1.4 and 1.5Correct

The arithmetic multiple-mixing estimates have the stated abstract hypotheses

PDF pages 5–7 and Section 4 · Theorems 1.4–1.5 and proofs · arXiv:1701.00945v2

The $S$ -algebraic statements assume explicit Sobolev properties and a quantitative two-point mixing bound. The proof uses only those inputs to obtain decay in the minimum pairwise height, first for ordinary correlations and then uniformly for diagonal-invariant couplings. The compact-open invariance and Sobolev-degree losses are retained in all quantified constants.

Final arXiv version 2 ↗

Theorem 1.6Correct

The adelic multiple-mixing conclusion is supported

PDF pages 7–8 and Section 5 · Theorem 1.6 and proof · arXiv:1701.00945v2

For a simply connected absolutely simple group that is isotropic at an Archimedean place, the proof divides according to whether the Archimedean separation is large. The large case applies the uniform $S$ -algebraic coupling estimate, while the complementary case invokes effective equidistribution of the relevant closed orbit; balancing the two errors gives the claimed decay in adelic height. The typographical word correction recorded in the proof section does not change this hypothesis.

Final arXiv version 2 ↗

02Proofs4 reported findingsCorrect

The coupling induction, its real and arithmetic specializations, and the adelic case split are correct and complete. Two uniquely determined notation errors are harmless and do not lower the overall proof status.

Sections 2 and 6Correct and complete

The coupling induction closes with compatible scale choices

PDF pages 9–20 and 36–42 · reduction to and proof of the general coupling estimate · arXiv:1701.00945v2

The argument partitions the indices into two clusters, averages along a root direction that separates the clusters, and decomposes the correlation error into slow-motion, two-point-mixing, and lower-order-coupling terms. Sobolev growth is polynomial in the averaging scale, whereas the available mixing term decays exponentially; the selected scale balances these terms. Induction on the number of factors then yields decay in the minimum pairwise separation without losing uniformity in the coupling.

Final arXiv version 2 ↗

Sections 4 and 5Correct and complete

The arithmetic and adelic reductions preserve all required uniformity

PDF pages 23–35 · arithmetic and adelic proofs · arXiv:1701.00945v2

The $S$ -algebraic proof verifies that height and adjoint growth provide the same three estimates used in the real coupling argument. In the adelic proof, the large-Archimedean branch uses that uniform estimate on a projected coupling and the small-Archimedean branch applies the cited effective closed-orbit equidistribution theorem with its isotropy hypothesis. The final choice of the splitting parameter makes both branches decay by a positive power of the minimum pairwise height.

Final arXiv version 2 ↗

Equations (1.2) and (1.3)Typo

Two matrix-coefficient inequalities omit modulus signs

PDF page 2 · equations (1.2)–(1.3) · arXiv:1701.00945v2

The left sides are printed as $\langle\pi(g)v_1,v_2\rangle$ in inequalities whose right sides are nonnegative real quantities. The unique correction is $|\langle\pi(g)v_1,v_2\rangle|$ in both displays. The subsequent correlation estimate is written with modulus, and every application uses the absolute-value form, so no argument or theorem changes.

Final arXiv version 2 ↗

Theorem 1.6 hypothesisTypo

The word “isotopic” should be “isotropic”

PDF page 7 · statement of Theorem 1.6 and following discussion · arXiv:1701.00945v2

The hypothesis twice says that the algebraic group is “isotopic” over an Archimedean completion. The unique mathematical term required by the cited effective-equidistribution input is “isotropic.” The surrounding discussion and the proof identify that same standard isotropy hypothesis, so the correction is mechanical and harmless.