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Bounded orbits of nonquasiunipotent flows on homogeneous spaces

Dmitry Y. Kleinbock, G. A. Margulis

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Journal citation: American Mathematical Society Translations 171, 141–172 (1996)
Publication date: 1996
DOI: 10.1090/trans2/171/11 ↗

Role in dependence graphs

Proof-critical source

Bounded trajectories of quasi-rays

This paper is included only for the following marked statements:

Lemma 1.4(a) · source printed p. 5Marstrand-type slicing lower bound.
Lemma 2.4.7(a–b) · source printed p. 14Gives Sobolev tensor-product control and normalized nonnegative smooth bump functions with norm $O(r^{-(\ell+N/2)})$ . The focal paper directly uses only bump existence; Shi uses the Sobolev control in Lemma 4.4 and in the bump constructed before (4.10).
Proposition 3.3 · source printed pp. 17–18Arbitrarily small tessellation domains in the connected simply connected nilpotent horospherical group.
Proposition 3.4 · source printed p. 18The tile-count estimate used in the Cantor construction.

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:

Corollary 2.4.4 · author PDF printed p. 13; Theorem 2.4.3 immediately precedingUniform exponential matrix-coefficient decay for a family of representations isolated from the trivial representation on every simple factor; this is BG23's EQ2 input.

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Author-hosted published text · American Mathematical Society Translations, Series 2, volume 171 (1996), pages 141–172

Dmitry Y. Kleinbock and G. A. Margulis. Bounded orbits of nonquasiunipotent flows on homogeneous spaces. American Mathematical Society Translations, Series 2, 171 (1996), 141–172.

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Generated August 23, 2026

01Statements3 reported findingsCorrect

The bounded-orbit theorem for nonquasiunipotent one-parameter subgroups, its representation-theoretic and geometric inputs, and the stated extensions are correct at their printed scope. The main construction has a formal branch-indexing defect, but the defect has a direct verified repair and does not change the theorem.

Theorem 1.1Correct

The bounded-orbit conclusion follows after a branch-dependent return-time repair

Author copy, printed pages 1–5 and 21–23 · Theorem 1.1 and Section 4.2

For a nonquasiunipotent one-parameter subgroup of a semisimple group acting on a finite-volume homogeneous space, the paper constructs a positive-Hausdorff-dimension set of points with bounded forward trajectories. The local tree construction, mixing estimate, and dimension calculation are valid. In the last compactness step, return times must be indexed by the marked translate chosen along each nested branch; with that correction, consecutive returns have uniformly bounded gaps and compact interpolation gives boundedness for every time.

Author-hosted published text ↗

Section 2Correct

The uniform mixing and smooth-partition inputs have the needed strength

Author copy, printed pages 8–15 · Theorem 2.4.3, Corollary 2.4.4, and Lemma 2.4.7

The paper isolates the representation-theoretic hypothesis required for uniform exponential matrix-coefficient decay and then derives the family-uniform estimate used in the counting argument. The smooth bump and product estimates retain their dimension and derivative losses. Corollary 2.4.4 is therefore a valid ambient exponential-mixing input, including for its later use in the dependence graph.

Author-hosted published text ↗

Sections 3 and 5Correct

The tessellation and extension statements preserve the hypotheses used in the proof

Author copy, printed pages 16–21 and 24–30 · tessellation construction and generalizations

The expanding-horospherical tessellation has the required bounded overlap, scale control, and boundary regularity. The later variants alter only the representation or invariant submanifold used to supply those same quantitative inputs; they do not silently remove a spectral or geometric hypothesis needed by the argument.

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02Proofs3 reported findingsContains incorrect or incomplete proofs

The analytic and geometric core is sound, but the final return-time recursion is not defined on the branch states created by the preceding line. A branch-dependent recursion repairs the proof without changing any estimate. One nearby symbol error is harmless.

Equations (4.4)–(4.8)Incomplete as written · verified repair

The return-time recursion ignores the marked branch

Author copy, printed page 22 · equations (4.4)–(4.8)

A child cylinder is defined using a marked translate $\gamma$ and the next state $\gamma g_{t(y)}y\in K$ . Equation (4.6), however, defines a single branch-independent quantity $t_{j+1}(y)=t(y)+t_j(g_{t(y)}y)$ . The unmarked point $g_{t(y)}y$ need not lie in $K$ , so the right side can be undefined; different children can also use different $\gamma$ . Consequently equation (4.7) does not follow from (4.4)–(4.6) as printed. For the repair, fix $h\in A_\infty(x)$ and choose a compatible infinite chain of nested cylinders, recording its marks $\gamma_j$ . Put $y_0=x$ , $y_{j+1}=\gamma_j g_{t(y_j)}y_j$ , and $\tau_{j+1}=\tau_j+t(y_j)$ . Unrolling the depth- $j$ cylinder representation leaves a residual element $v_j\in V$ and gives $g_{\tau_j}hx=v_jy_j$ . The final mark in that depth- $j$ representation was selected precisely so that $Vy_j\subset K$ , hence every displayed return lies in $K$ . Since all increments lie in $[T,T']$ , compact interpolation between successive returns proves (4.8).

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Equation (4.4) paragraphTypo

The density function has one wrong argument

Author copy, printed page 22 · paragraph preceding equation (4.4)

The density is written once as $\delta(y,k,t(y))$ , although the compact target throughout the construction is $K$ . The unique correction is $\delta(y,K,t(y))$ . The same paragraph and all estimates use the compact set $K$ , so the typo has no mathematical effect.

Author-hosted published text ↗

Sections 2–4 and AppendixCorrect and complete

The remaining quantitative estimates and dimension argument close

Author copy, printed pages 8–23 and 30–32

Exponential mixing controls the proportion of admissible children, boundary estimates control losses near partition walls, and the mass-distribution calculation yields the announced lower bound on Hausdorff dimension. The appendix supplies the required representation-theoretic decay at exactly the uniformity used earlier. None of these steps depends on the defective branch-independent notation in (4.6).