The Dirichlet spectrum

The focal proofs use Kleinbock–Weiss Theorem 2.2 to show that the maximal-limsup set has full measure for every weighted diagonal ray. This supplies the dense large-value set in the topological interpolation theorem. It supports Corollary 1 only through Theorem 3 and does not repair the endpoint exact-exponent gap.

• Theorem 1The full Dirichlet spectrum $[0,\Delta]$ and dense uncountability of every level set when $\max(m,n)>1$.
• Theorem 2The extended Dirichlet spectrum of every continuous decreasing compact exhaustion of $\mathcal{X}_d$ is the full interval $[0,b]$, with dense uncountable horospherical level sets.
• Theorem 3The $\psi$-Dirichlet spectrum is $[0,\infty]$ under the printed decay hypotheses, with dense uncountable level sets.
• Corollary 1Claims dense uncountably many matrices of exact uniform exponent $\gamma$ at every critical limsup $c\in[0,\infty]$.

The proof of focal Theorem 4 needs a fixed compact subset of $\mathcal{X}_2$ met infinitely often by every nondivergent orbit; the cited proof of Lemma 11.29 supplies it.

• Theorem 4The analogous extended spectrum statement in dimension two for arbitrary lattices.

Kleinbock–Weiss uses Shah's quantitative recurrence, linearization theorem, and unipotent-box projection lemma to prove tightness and exclude concentration on a proper homogeneous stratum.

• Theorem 2.2Uniform equidistribution of expanding matrix-horosphere averages along every ray drifting away from Weyl-chamber walls.

Dani Theorem 9.1 classifies finite ergodic measures invariant under the matrix horosphere, giving Kleinbock–Weiss Proposition 2.7.

• Theorem 2.2Uniform equidistribution of expanding matrix-horosphere averages along every ray drifting away from Weyl-chamber walls.

Claim 3.2 gives invariance of weak limits under the expanding matrix horosphere.

• Theorem 2.2Uniform equidistribution of expanding matrix-horosphere averages along every ray drifting away from Weyl-chamber walls.

Highest-weight decomposition is used in Lemma 2.3 to prove exponential growth of every nonfixed representation vector throughout the positive cone.

• Theorem 2.2Uniform equidistribution of expanding matrix-horosphere averages along every ray drifting away from Weyl-chamber walls.

Dani Lemma 2.4 supplies the rank-one case of Shah's quantitative return alternative.

• Theorem 2.2Quantitative compact-return alternative for bounded unipotent boxes in arbitrary finite-volume homogeneous spaces.

After arithmeticity, the final EMS Theorem 3.9 supplies the higher-rank non-divergence alternative.

• Theorem 2.2Quantitative compact-return alternative for bounded unipotent boxes in arbitrary finite-volume homogeneous spaces.

Shah 1994 Theorem 5.2 and Proposition 5.4 yield the linearization alternative restated as Shah 1996 Theorem 4.1.

• Theorem 4.1Linearization alternative near a proper homogeneous stratum.

The book supplies lattice, solvable-radical, and compact-factor reductions preceding Shah's rank split.

• Theorem 2.2Quantitative compact-return alternative for bounded unipotent boxes in arbitrary finite-volume homogeneous spaces.

Arithmeticity sends the irreducible higher-rank lattice branch into the arithmetic setting required by EMS.

• Theorem 2.2Quantitative compact-return alternative for bounded unipotent boxes in arbitrary finite-volume homogeneous spaces.

Dani 1981's finite-measure classification uses the earlier Haar characterization and invariant-measure machinery from Theorem 2.4 and Proposition 9.3.

• Theorem 9.1Classification of finite ergodic measures invariant under a maximal horospherical subgroup.

Dani 1981 Proposition 3.1 explicitly imports the Garland–Raghunathan construction of fundamental domains for nonuniform real-rank-one lattices. Its finite cusp covering and horospherical lattice property feed Theorems 3.4 and 8.2 and ultimately the selected Theorem 9.1.

• Theorem 9.1Classification of finite ergodic measures invariant under a maximal horospherical subgroup.

The accessible Garland–Raghunathan proof indication says Theorem 1.2 relies on Kazhdan–Margulis and that its central Lemmas 3.3 and 3.5 use those ideas. Those lemmas imply the compact ray-quotient Lemma 3.2 and hence the finite-cusp covering and cusp-lattice assertions selected here.

• Theorem 0.6Finite rank-one cusp representatives, lattice intersections in the corresponding horospherical groups, and a finite Siegel-set covering.

Canonical local coordinates and the small-neighborhood commutator estimate yield Kazhdan–Margulis Lemma 2, which places every sufficiently small part of a discrete subgroup in a connected nilpotent subgroup.

• Theorem 1 and corollaryUniform bounded conjugation of sufficiently small elements in arbitrary discrete subgroups, and the resulting positive lower bound for quotient covolume.

Borel–Tits Sections 2.3 and 4.15–4.16 and Lemma 8.3/Proposition 8.4 put the connected unipotent radical $N_u$ inside the positive-root unipotent subgroup $U^+(T_0)$, the exact containment used in Kazhdan–Margulis Lemma 3.

• Theorem 1 and corollaryUniform bounded conjugation of sufficiently small elements in arbitrary discrete subgroups, and the resulting positive lower bound for quotient covolume.

Borel Theorem 11.1 supplies the connected nilpotent algebraic-group factorization $N=N_s imes N_u$ used twice at the start of Kazhdan–Margulis Lemma 3.

• Theorem 1 and corollaryUniform bounded conjugation of sufficiently small elements in arbitrary discrete subgroups, and the resulting positive lower bound for quotient covolume.

Kazhdan–Margulis cites Rosenlicht for the assertion that the relevant intersection-dimension function is locally generically minimal. Exhaustive inspection of the cited article found no such theorem; the edge records the historical citation without treating it as a valid proof.

• Theorem 1 and corollaryUniform bounded conjugation of sufficiently small elements in arbitrary discrete subgroups, and the resulting positive lower bound for quotient covolume.

The standard fiber-dimension theorem makes the minimum-dimensional torus intersections persist on a dense open set and therefore supplies the exact local inequality needed in Lemma 3. This is an explicitly disclosed repair, not a relabeling of the insufficient Rosenlicht citation.

• Theorem 1 and corollaryUniform bounded conjugation of sufficiently small elements in arbitrary discrete subgroups, and the resulting positive lower bound for quotient covolume.

Rosenlicht's rational fixed-point lemma, splitness corollary, and triangularization proposition give the rational stable flag and fixed point used in Borel–Tits Sections 2.7 and 8.1–8.2.

• Theorem 8.2Supplies the rational split-solvable and fixed-point mechanism supporting the positive-root construction.
• Lemma 8.3 and Proposition 8.4Directly yields containment of the connected unipotent subgroup in the positive-root unipotent group required in Kazhdan–Margulis Lemma 3.

Exposé 13 Theorem 1(d) gives the closed iterated root-subgroup decomposition used verbatim in Borel–Tits Section 2.3.

• Section 2.3 root-subgroup structureDecomposes a torus-normalized subgroup of the positive unipotent radical into its root subgroups.

Jacobson–Morozov supplies the three-dimensional split simple Lie subalgebra attached to a nilpotent element in the characteristic-zero argument.

• Theorem 4.15 and Corollary 4.16Places a unipotent subgroup in a parabolic and then in a suitable unipotent radical.

The reductive-group existence and centralizer facts cited from SGA 3 support the parabolic and split-torus construction in the selected chain.

• Theorem 4.15 and Corollary 4.16Places a unipotent subgroup in a parabolic and then in a suitable unipotent radical.
• Theorem 8.2Supplies the rational split-solvable and fixed-point mechanism supporting the positive-root construction.

Borel's later monograph independently proves both selected claims: Jordan factors stay in the group and a connected nilpotent algebraic group splits as its torus times its unipotent subgroup.

• Theorem 8.4Jordan semisimple and unipotent factors remain inside the algebraic group.
• Theorem 11.1A connected nilpotent algebraic group is the direct product of its semisimple torus and unipotent subgroup, as used in Kazhdan–Margulis Lemma 3.

Chevalley Theorem 18 independently proves the characteristic-zero Jordan-decomposition statement required downstream.

• Theorem 8.4Jordan semisimple and unipotent factors remain inside the algebraic group.

Rosenlicht explicitly imports Borel's presentation of Lie–Kolchin triangularization and follows Borel's fixed-point proof pattern.

• Corollary 1Triangularizes a unipotent matrix group over the ground field when every component is defined there.
• Rational fixed-point lemmaA split connected solvable group acting on a complete variety with a rational point has a rational fixed point.
• Proposition 7Triangularizes a connected split solvable matrix group over the ground field.

Theorem 5 and its corollary give the quotient action and ground-field descent used in the fixed-point induction.

• Rational fixed-point lemmaA split connected solvable group acting on a complete variety with a rational point has a rational fixed point.

Weil's final theorem replaces the prehomogeneous rational quotient by a birational homogeneous-space model, the exact regularization Rosenlicht uses before compactifying the one-dimensional orbit.

• Rational fixed-point lemmaA split connected solvable group acting on a complete variety with a rational point has a rational fixed point.

Rosenlicht Theorem 5 uses the quotient group constructed in its internal Theorem 4, whose direct proof invokes Weil's homogeneous-space regularization theorem.

• Theorem 5 and corollaryConstructs the quotient action by a normal subgroup and descends it to the ground field for Rosenlicht's 1957 fixed-point induction.

The open-subset consequence produces a rational point over every sufficiently large finite extension; Weil chooses the extension degree prime and larger than the fixed cycle degree in Proposition 9's finite-field branch.

• Final unnumbered theorem, clause (ii)Replaces a prehomogeneous rational action by a birationally equivalent homogeneous-space model over the ground field.

Weil 1955 Appendix Proposition 7 directly cites Weil 1954 Lemma 7 for the specialization/open-set step; this dependency is independent of the Lang–Weil route.

• Proposition 7 and Appendix inputThe open-set and specialization construction used in the descent argument imports Weil 1954 Lemma 7 directly, independently of the Lang–Weil route.

Weil's cycle, specialization, abstract-variety, and descent machinery supplies the terminal foundational results used throughout the selected construction.

• Final unnumbered theorem, clause (ii)Replaces a prehomogeneous rational action by a birationally equivalent homogeneous-space model over the ground field.
• Proposition 7 and Appendix inputThe open-set and specialization construction used in the descent argument imports Weil 1954 Lemma 7 directly, independently of the Lang–Weil route.

The monograph supplies the symmetric-function and rational-cycle descent input in the selected construction.

• Final unnumbered theorem, clause (ii)Replaces a prehomogeneous rational action by a birationally equivalent homogeneous-space model over the ground field.

Section I.6 and Lemma 4 show that the bad-hyperplane locus and the non-variety Chow locus are algebraic, descended, and proper, enabling the hyperplane-section induction.

• Theorem 1Bounds the rational-point count of a projective variety over a finite field.
• Open-subset extension-field consequenceGives a rational point on a nonempty open subset over every sufficiently large finite extension, as required in Weil 1955.

Cycle terminology, Bézout, and foundational intersection results supply the book-level inputs in Lang–Weil's induction.

• Theorem 1Bounds the rational-point count of a projective variety over a finite field.

The function-field Riemann-hypothesis estimate supplies the dimension-one base case of Theorem 1.

• Theorem 1Bounds the rational-point count of a projective variety over a finite field.

Foundations supplies the multiplicity, generic-intersection, regular-extension, intersection-associativity, cycle, and specialization results invoked in the selected lemmas.

• Section I.6 Chow-locus constructionShows that reducible and multiple-component cycle loci are closed and defined over the prime field.
• Lemma 4Makes the bad-hyperplane locus algebraic, descended to every field of definition, and of dimension strictly below the dual projective space.
• Lemma 7Supplies the specialization/open-set step directly imported by Weil 1955 Appendix Proposition 7.

The Galois-theoretic intermediate-field theorem closes the generic-hyperplane irreducibility argument that supports Lemma 4.

• Lemma 4Makes the bad-hyperplane locus algebraic, descended to every field of definition, and of dimension strictly below the dual projective space.

Dani 1981 Proposition 8.1 imports Raghunathan Theorem 5.22 for the finite product decomposition used at the start of Theorem 8.2. The distinct Garland–Raghunathan article cited by Proposition 3.1 is represented by its own graph node and is not folded into this book edge.

• Theorem 9.1Classification of finite ergodic measures invariant under a maximal horospherical subgroup.

Dani's proof of Proposition 3.1 and the higher-rank branch of Theorem 8.2 invoke Margulis arithmeticity to reduce irreducible nonuniform lattices to the arithmetic setting.

• Theorem 9.1Classification of finite ergodic measures invariant under a maximal horospherical subgroup.

The rank-one case in Dani Theorem 8.2 invokes Warner's Bruhat decomposition theorem to split the open and exceptional cells used in the measure argument.

• Theorem 9.1Classification of finite ergodic measures invariant under a maximal horospherical subgroup.

Dani Proposition 3.1 explicitly uses Borel reduction theory for the arithmetic cusp decomposition that feeds Theorems 3.4, 8.2, and ultimately 9.1.

• Theorem 9.1Classification of finite ergodic measures invariant under a maximal horospherical subgroup.

Shah Theorem 3.1 explicitly imports Dani–Margulis Theorem 6.1, and the Section 5 linearization induction uses the same chapter's quantitative polynomial-trajectory tools.

• Theorem 3.1Quantitative non-divergence for polynomial trajectories.
• Theorem 5.2 and Proposition 5.4Linearization near a proper homogeneous stratum and the one-parameter base case.

Mozes–Shah Proposition 3.4 is the one-parameter base used in Shah Proposition 5.4.

• Theorem 5.2 and Proposition 5.4Linearization near a proper homogeneous stratum and the one-parameter base case.

Mozes–Shah states that Proposition 3.4 follows from Dani–Margulis Theorem 7.3; its shorter proof uses two supporting propositions from the same chapter.

• Proposition 3.4Uniform stabilizer-or-small-time alternative near a compact part of a homogeneous stratum.

EMS Theorem 3.4 extends the Dani–Margulis Theorem 6.1 polynomial non-divergence argument to its exponential-polynomial class.

• Theorem 3.4Quantitative measure recurrence for the bounded exponential-polynomial trajectory class.

Dani–Margulis Proposition 1.8 is restated in EMS notation as Proposition 3.7 and closes the compactness step in Theorem 3.5.

• Proposition 3.7Reduction-theoretic compactness criterion inside Theorem 3.5.
• Theorems 3.5 and 3.9Arithmetic parabolic alternative and the combined higher-rank non-divergence theorem.

Arithmetic reduction theory supplies the rational parabolic and compactness framework for the finite induction in EMS Theorem 3.5.

• Theorems 3.5 and 3.9Arithmetic parabolic alternative and the combined higher-rank non-divergence theorem.

Borel reduction theory gives the finite cusp decomposition and compactness criterion used in the setup and proof of Proposition 1.8.

• Proposition 1.8Compactness of the quotient set defined by an admissible parabolic sequence and two-sided reduction-function bounds.

Rank-one lattice and structural facts reduce Dani's exterior-power cusp argument to the stated setting.

• Lemma 2.4Rank-one compact-return alternative used in Shah 1996 Theorem 2.2.

Arithmetic-group reduction and cusp decomposition are used in Dani's selected invariant-measure branch.

• Theorem 2.4 and Proposition 9.3Haar-measure characterization and the invariant-measure step used by Dani 1981 Theorem 9.1.

The book supplies lattice and homogeneous-space structure used in Dani's selected branch.

• Theorem 2.4 and Proposition 9.3Haar-measure characterization and the invariant-measure step used by Dani 1981 Theorem 9.1.

Semisimple lattice structure and arithmetic facts enter the selected horospherical-measure classification route.

• Theorem 2.4 and Proposition 9.3Haar-measure characterization and the invariant-measure step used by Dani 1981 Theorem 9.1.

Moore ergodicity and associated homogeneous-space facts support Dani's selected measure-classification branch.

• Theorem 2.4 and Proposition 9.3Haar-measure characterization and the invariant-measure step used by Dani 1981 Theorem 9.1.

Lie-theoretic decomposition and Haar-measure facts are terminal ingredients in Dani's selected branch.

• Theorem 2.4 and Proposition 9.3Haar-measure characterization and the invariant-measure step used by Dani 1981 Theorem 9.1.