Bounded trajectories of quasi-rays

The focal paper uses the horospherical–parabolic identification to construct the parabolic $P$ and maximal connected Ad-diagonalizable subgroup $A$ on which every subsequent cone and equidistribution argument is built.

• Theorem 1.4Thickness of bounded quasi-ray trajectories on the homogeneous space and along the chosen horospherical subgroup.
• Theorem 1.5Thickness of badly approximable matrices for quasimultiplicative weight functions.

Shi's expanding-cone description fixes the admissible cone, and Theorem 1.5 specialized to $k=1$ is exactly the focal paper's Theorem 2.5 exponential $U$-slice estimate. The focal chain then runs through Corollary 2.7, Propositions 3.2 and 3.6, and Theorem 3.1 to the headline thickness result.

• Theorem 1.4Thickness of bounded quasi-ray trajectories on the homogeneous space and along the chosen horospherical subgroup.
• Theorem 1.5Thickness of badly approximable matrices for quasimultiplicative weight functions.

The strong spectral-gap statement discharges a hypothesis of Shi's effective equidistribution theorem for the irreducible lattice quotient used here.

• Theorem 1.4Thickness of bounded quasi-ray trajectories on the homogeneous space and along the chosen horospherical subgroup.
• Theorem 1.5Thickness of badly approximable matrices for quasimultiplicative weight functions.

Shah's representation-theoretic expansion lemma implies that the super-expanding cone is nonempty and contains an admissible one-parameter ray, which is needed to put Shi's estimate into the form used by Corollary 2.7.

• Theorem 1.4Thickness of bounded quasi-ray trajectories on the homogeneous space and along the chosen horospherical subgroup.
• Theorem 1.5Thickness of badly approximable matrices for quasimultiplicative weight functions.

Cone duality identifies distance to the boundary with the minimum positive dual functional, allowing the wall-distance parameter in Shi's theorem to become the geometric quantity used throughout the quasi-ray construction.

• Theorem 1.4Thickness of bounded quasi-ray trajectories on the homogeneous space and along the chosen horospherical subgroup.
• Theorem 1.5Thickness of badly approximable matrices for quasimultiplicative weight functions.

The restricted-root decomposition drives the contraction and Jacobian estimates in the main Cantor argument. Further root-system facts support the separately marked appendix propositions and rank-two example.

• Theorem 1.4Thickness of bounded quasi-ray trajectories on the homogeneous space and along the chosen horospherical subgroup.
• Theorem 1.5Thickness of badly approximable matrices for quasimultiplicative weight functions.
• Proposition A.4An appendix criterion comparing algebraically and geometrically expanding cones for abelian horospherical subgroups.
• Proposition A.5An appendix minimality consequence of the cone inclusion.
• Example A.8The rank-two $B_2/C_2$ cone comparison.

The focal proof imports slicing, bump-function existence, nilpotent tessellations, and the tile-count estimate. Those results transfer the horospherical dimension bound to the homogeneous space and make the strongly tree-like construction quantitative.

• Theorem 1.4Thickness of bounded quasi-ray trajectories on the homogeneous space and along the chosen horospherical subgroup.
• Theorem 1.5Thickness of badly approximable matrices for quasimultiplicative weight functions.

Urbański's tree-like dimension estimate is the final dimension engine: retained densities and exponentially shrinking diameters yield the full local dimension required by Theorem 3.1.

• Theorem 1.4Thickness of bounded quasi-ray trajectories on the homogeneous space and along the chosen horospherical subgroup.
• Theorem 1.5Thickness of badly approximable matrices for quasimultiplicative weight functions.

Mahler's compactness criterion converts boundedness of the lattice trajectory into a uniform lower bound on integral vectors, which is exactly the bridge used in Proposition 4.4's generalized Dani correspondence.

• Theorem 1.5Thickness of badly approximable matrices for quasimultiplicative weight functions.

The focal paper explicitly presents its complementary zero-full Lebesgue-measure statement as a direct corollary of Kleinbock–Wang's rectangular limsup theorem. It does not feed either headline theorem.

• Theorem 4.1The complementary zero-full Lebesgue-measure law recalled as a direct corollary of an earlier theorem.

The root-subspace bracket theorem proves Appendix Lemma A.6, which is then used to establish the abelian cone-inclusion criterion in Proposition A.4. The focal paper states that this appendix is illustrative and not needed for its main results.

• Proposition A.4An appendix criterion comparing algebraically and geometrically expanding cones for abelian horospherical subgroups.

The rank-two Cartan data determines the root inner-product signs used in the appendix's $B_2/C_2$ and $G_2$ examples.

• Example A.8The rank-two $B_2/C_2$ cone comparison.
• Example A.9The $G_2$ counterexample for cone inclusion.

Shah's Lemma 5.2 shows that uniformly nonzero vectors fixed by the expanding horospherical subgroup are expanded along the chamber-divergent ray. Shi uses it to infer from the simple-factor and horospherical hypotheses that $\{b_t:t>0\}\subset A_U^+$, making the depth parameter in (1.9) nontrivial.

• Theorem 1.5 ($k=1$)Effective equidistribution of a horospherical slice. For the focal specialization, the only analytic branches are quantitative nonescape at (4.16) and spectral-gap thickening at (4.18); the later induction steps for multiple correlations are outside the marked scope.

Exponential coordinates identify the simply connected nilpotent group $U$ with $\mathfrak u$ and Haar measure with normalized Lebesgue measure. Shi needs this conversion for the polynomial-good estimates in Lemma 3.2 and the polar-coordinate reduction in Lemma 3.3.

• Theorem 1.3Quantitative nonescape of mass for expanding $U$-slices. It bounds the bad slice set in (4.16), one of the two proof-critical inputs to the $k=1$ case of Theorem 1.5.

Proposition 3.5 converts the shortest-vector compactum $K_\varepsilon$ into a quantitative injectivity-radius compactum. This lets Shi's estimate (3.5) imply the injectivity-radius formulation required by Theorem 1.3.

• Theorem 1.3Quantitative nonescape of mass for expanding $U$-slices. It bounds the bad slice set in (4.16), one of the two proof-critical inputs to the $k=1$ case of Theorem 1.5.

The polynomial-good lemma applies uniformly to the squared exterior-power covolume functions $\psi_{a,v}^2$, verifying the good-function hypothesis in Shi's arithmetic-factor nondivergence argument.

• Theorem 1.3Quantitative nonescape of mass for expanding $U$-slices. It bounds the bad slice set in (4.16), one of the two proof-critical inputs to the $k=1$ case of Theorem 1.5.

Kleinbock's Theorem 2.2 applies once Shi has checked good covolume functions and uniform lower bounds, yielding (3.5), the $O(\varepsilon^\delta)$ bad-set estimate for every arithmetic factor.

• Theorem 1.3Quantitative nonescape of mass for expanding $U$-slices. It bounds the bad slice set in (4.16), one of the two proof-critical inputs to the $k=1$ case of Theorem 1.5.

The one-variable polynomial estimate and the multivariable derivative induction prove the polynomial-good lemma that Bernik–Kleinbock–Margulis state as Lemma 3.2 and that Shi later imports.

• Lemma 3.2Shows that every degree-$\leq l$ polynomial on $\mathbb{R}^d$ is $(C_{d,l},1/(dl))$-good, verifying the good-function hypothesis in Shi's arithmetic nondivergence branch.

Kleinbock–Weiss Proposition 3.1 supplies the finite cusp-vector set with discreteness, the precompactness criterion, and uniqueness of a sufficiently short vector. Shi quotes it as Lemma 3.4 and uses it in Lemma 3.5 and the rank-one proof of Lemma 3.3.

• Theorem 1.3Quantitative nonescape of mass for expanding $U$-slices. It bounds the bad slice set in (4.16), one of the two proof-critical inputs to the $k=1$ case of Theorem 1.5.

The finite-cusp reduction decomposition, cusp lattices, and overlap-separation property prove the compactness equivalence and uniqueness assertions in Kleinbock–Weiss Proposition 3.1.

• Proposition 3.1Constructs a finite cusp-vector family satisfying discreteness, a precompactness criterion, and uniqueness of a sufficiently short vector; Shi quotes it as Lemma 3.4.

The Dani–Margulis discreteness theorem supplies part (1) of Kleinbock–Weiss Proposition 3.1 for each cusp-vector orbit.

• Proposition 3.1Constructs a finite cusp-vector family satisfying discreteness, a precompactness criterion, and uniqueness of a sufficiently short vector; Shi quotes it as Lemma 3.4.

Buenger–Zheng's explicit rank-one unipotent nondivergence estimate controls each one-dimensional slice. Shi explicitly needs the calculated dependence of $c_\varepsilon$, then integrates (3.10) in polar coordinates to obtain (3.9).

• Theorem 1.3Quantitative nonescape of mass for expanding $U$-slices. It bounds the bad slice set in (4.16), one of the two proof-critical inputs to the $k=1$ case of Theorem 1.5.

The real-rank-one Bruhat decomposition and existence of a Zassenhaus neighborhood let Buenger–Zheng classify the small discrete nilpotent pieces into conjugates of $MA$ or $MN$, the structural basis of their interval linearization.

• Theorem 1.1Explicit one-parameter unipotent nondivergence on a rank-one quotient; Shi applies its explicit $c_\varepsilon$ calculation in (3.10) and integrates in polar coordinates.

The good-function and quantitative-nondivergence results, including the arbitrary-discrete-lattice version, give Buenger–Zheng Theorem 4.1 and Corollary 4.1. Those estimates measure the bad times in the proof of their Theorem 1.1.

• Theorem 1.1Explicit one-parameter unipotent nondivergence on a rank-one quotient; Shi applies its explicit $c_\varepsilon$ calculation in (3.10) and integrates in polar coordinates.

The Borel-density consequence that $Z_G\Gamma$ is discrete makes the center quotient finite, allowing Shi's Lemma 3.6 reduction to arithmetic and real-rank-one factors. The surrounding product/arithmeticity decomposition is invoked without a source and is therefore noted but not represented by an invented edge.

• Theorem 1.3Quantitative nonescape of mass for expanding $U$-slices. It bounds the bad slice set in (4.16), one of the two proof-critical inputs to the $k=1$ case of Theorem 1.5.

The stable-central-unstable multiplication theorem for a class-$A$ element gives Shi's open conull product cell and Haar integration formula (4.1), the geometry used to thicken a $G^+$-slice.

• Lemma 4.1 and Haar decomposition (4.1)Provides the stable-central-unstable conull cell and integration formula used to thicken a $G^+$-slice.

EMV convert spectral gap into a temperedness exponent and a Sobolev matrix-coefficient bound. Because every simple-factor projection of $b$ is nontrivial, their product-group estimate decays exponentially along $b_t$, exactly the conclusion of Shi Lemma 4.2.

• Lemma 4.2Turns the assumed spectral gap into exponential Sobolev matrix-coefficient decay along $b_t$ when every simple-factor projection of $b$ is nontrivial.

Lemma 2.4.7 supplies the normalized bump and Sobolev loss used in Shi Lemma 4.4 and in the analogous $G^+$ bump before (4.10). This quantitative loss is balanced against nonescape and mixing to obtain (4.18).

• Lemmas 4.3–4.4Thickens a good $G^+$-slice with a controlled bump function and combines it with Lemma 4.2 to produce the good-slice estimate (4.18).

The $L^{2+\varepsilon}$ characterization of temperedness and Harish-Chandra majorization convert temperedness into the explicit matrix-coefficient estimates (6.7)–(6.9).

• (6.1), (6.3), (6.9), and (6.10)Converts spectral gap to a uniform temperedness exponent and then to Sobolev matrix-coefficient decay, including the product-group weak-$\varphi_0$ estimate used in Shi Lemma 4.2.

Cowling's integrability theorems prove spectral gap implies a uniform finite $L^p$ exponent for higher-rank simple groups and rank-one groups with property $(T)$, leaving only $SO(n,1)$ and $SU(n,1)$ for EMV's complementary-series argument.

• (6.1), (6.3), (6.9), and (6.10)Converts spectral gap to a uniform temperedness exponent and then to Sobolev matrix-coefficient decay, including the product-group weak-$\varphi_0$ estimate used in Shi Lemma 4.2.

Knapp's spherical-function bound gives EMV (6.3); his matrix-coefficient asymptotics and leading-exponent result control the remaining rank-one Langlands quotients in Appendix C.

• (6.1), (6.3), (6.9), and (6.10)Converts spectral gap to a uniform temperedness exponent and then to Sobolev matrix-coefficient decay, including the product-group weak-$\varphi_0$ estimate used in Shi Lemma 4.2.

The critical-abscissa and endpoint-unitarity results place every nontrivial complementary-series endpoint in a discrete subset of $[0,1)$, yielding the uniform finite temperedness exponent required for EMV (6.1).

• (6.1), (6.3), (6.9), and (6.10)Converts spectral gap to a uniform temperedness exponent and then to Sobolev matrix-coefficient decay, including the product-group weak-$\varphi_0$ estimate used in Shi Lemma 4.2.

Howe–Moore vanishing rules out a nontrivial unitary Langlands quotient at endpoint $z=1$, closing EMV's rank-one proof that spectral gap implies a finite temperedness exponent.

• (6.1), (6.3), (6.9), and (6.10)Converts spectral gap to a uniform temperedness exponent and then to Sobolev matrix-coefficient decay, including the product-group weak-$\varphi_0$ estimate used in Shi Lemma 4.2.

Kleinbock–Margulis cite Marstrand's slicing theorem for the dimension lower bound stated as Lemma 1.4(a). Falconer's later book is an alternative source, not a second mandatory prerequisite.

• Lemma 1.4(a)Marstrand-type slicing lower bound.

The strong Malcev basis supplies the nested ideals used in the internal induction that constructs arbitrarily small nilpotent tessellation domains. The remaining scaled-cube proof, including the monotone choice used by the focal paper, is internal.

• Proposition 3.3Arbitrarily small tessellation domains in the connected simply connected nilpotent horospherical group.

Wallach's result normalizes a nonzero restricted-root vector into an $\mathfrak{sl}_2$ triple, the first external input in Kraljević's proof.

• Root-subspace bracket theoremDetermines the bracket of restricted-root subspaces and is used in Appendix Lemma A.6.

Finite-dimensional $\mathfrak{sl}_2$ representation theory gives the integral weight decomposition and surjectivity of the raising map used to prove the root-subspace bracket equality.

• Root-subspace bracket theoremDetermines the bracket of restricted-root subspaces and is used in Appendix Lemma A.6.

Cowling's theorems provide a finite group-dependent matrix-coefficient integrability exponent for the property-(T), real-rank-at-least-two branch of Kelmer–Sarnak's general spectral-gap discussion.

• Property-(T) spectral-gap branchCombines property (T) with a finite group-dependent matrix-coefficient integrability exponent when the simple group has real rank at least two.

Arithmeticity identifies the irreducible lattice in the product case as arithmetic, allowing the congruence spectral-gap machinery to apply.

• Arithmetic product-lattice spectral-gap branchCombines arithmeticity, congruence spectral gap, automorphic transfer, and finite-cover invariance for irreducible lattices in products.

Burger–Sarnak's automorphic restriction theorem supplies the transfer mechanism in the congruence-lattice spectral-gap argument; Clozel supplies the complementary general property-$\tau$ input.

• Arithmetic product-lattice spectral-gap branchCombines arithmeticity, congruence spectral gap, automorphic transfer, and finite-cover invariance for irreducible lattices in products.

Clozel's proof of the $\tau$ conjecture completes the uniform congruence spectral gap invoked for the arithmetic product-lattice branch.

• Arithmetic product-lattice spectral-gap branchCombines arithmeticity, congruence spectral gap, automorphic transfer, and finite-cover invariance for irreducible lattices in products.

The finite-cover lemma passes spectral gap from a congruence lattice to the commensurable lattice under consideration. Its proof is internal, so Furman–Shalom, Rosenblatt, and Schmidt are excluded as contextual antecedents.

• Arithmetic product-lattice spectral-gap branchCombines arithmeticity, congruence spectral gap, automorphic transfer, and finite-cover invariance for irreducible lattices in products.

Finite generation of the $S$-arithmetic group permits Burger–Sarnak to choose the finite averaging operator used in Lemmas 2.1–2.2.

• Theorem 1.1(a)Transfers automorphic support under restriction to a semisimple rational subgroup.

Strong approximation makes the simply connected cover's $S$-integral points dense in the relevant real group in Burger–Sarnak Lemma 2.1.

• Theorem 1.1(a)Transfers automorphic support under restriction to a semisimple rational subgroup.

Weak approximation passes from the simply connected cover and its components to the algebraic group in Burger–Sarnak's proof.

• Theorem 1.1(a)Transfers automorphic support under restriction to a semisimple rational subgroup.

Fell topology and weak containment supply the diagonal matrix-coefficient characterization used in the final restriction argument.

• Theorem 1.1(a)Transfers automorphic support under restriction to a semisimple rational subgroup.

Burger–Sarnak's restriction and transfer theorem is structurally essential to Clozel's reduction to rank-one cases.

• Theorem 3.1Completes the congruence spectral-gap/property-$\tau$ input used in the general lattice argument.

The automorphic lift supplies the $\mathrm{SL}_2$ base case in Clozel's isotropic reduction. The remaining rank-one cases are proved within Clozel and receive no speculative inherited edges.

• Theorem 3.1Completes the congruence spectral-gap/property-$\tau$ input used in the general lattice argument.

The second-moment lower bound for a finite union is applied locally to the rectangular events. It yields positive limsup measure from the divergence and intersection estimates; the doubling-measure density step then upgrades this to full measure.

• Theorem 2.5Turns rectangular ubiquity, regularity, and divergence of the volume sum into full measure. Theorem 2.7 applies it after Lemma 6.2 constructs the needed ubiquitous system.

BDV Proposition 1 upgrades a uniform positive-density estimate in every sufficiently small ball to full measure. Kleinbock–Wang reproduce it as Lemma 3.2 and apply it after the local Chung–Erdős estimate.

• Theorem 2.5Turns rectangular ubiquity, regularity, and divergence of the volume sum into full measure. Theorem 2.7 applies it after Lemma 6.2 constructs the needed ubiquitous system.

The metric 5r covering theorem supplies the selection idea for Kleinbock–Wang's aligned-rectangle Lemma 3.3. The resulting disjoint subfamily and fivefold cover are used in the first two steps of Theorem 2.5.

• Theorem 2.5Turns rectangular ubiquity, regularity, and divergence of the volume sum into full measure. Theorem 2.7 applies it after Lemma 6.2 constructs the needed ubiquitous system.

Minkowski's convex-body theorem first handles the full-measure reduction when the volume product exceeds one, then produces a nonzero integer vector satisfying the simultaneous rectangular inequalities in Lemma 6.2. This establishes the ubiquity input for Theorem 2.5 and hence the divergence half of Theorem 2.7.

• Lemma 6.2Constructs the ubiquitous rectangular system used in the proof of Theorem 2.7; Minkowski's theorem produces its initial nonzero integer vector.
• Theorem 2.7The rectangular zero-full Lebesgue-measure law from which the focal paper's side theorem follows. Its divergence half runs through Lemmas 6.1–6.2 and Theorem 2.5; its convergence half is elementary Borel–Cantelli.

Federer's adequate-family selection theorem and inner regularity provide the disjoint covering and closed approximation used in BDV Lemmas 6–7; those lemmas prove Proposition 1's full-measure criterion.

• Proposition 1A uniform positive-density lower bound in every sufficiently small ball forces a Borel set to have full measure for a finite doubling measure. Kleinbock–Wang reproduce it as Lemma 3.2.