Einsiedler 2012 distribution closed geodesics modular surface duke

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proposition_3_6: Linnik's basic lemma in the paper's geometric form: the product measure of the pairs of points of the discriminant-d orbits below height H that lie within distance delta of each other is at most a constant times H^4 delta^3 d^epsilon, for d^(-1/4) <= delta <= H^(-2)/3.

proposition_4_3: The paper's covering estimate for the cusp: the points whose geodesic trajectory between times -N and N starts and ends below height M and lies above height M exactly at the times of a set V are covered by a constant times e^(2N - |V|/2) Bowen N-balls, and only a constant times e^((2 log log M / log M) N) sets V occur.

theorem_1_2: Skubenko's theorem as the paper recalls it: for a fixed prime p > 2, as d tends to infinity through the positive discriminants with (d/p) = 1, the scaled primitive forms of discriminant d equidistribute on the hyperboloid b^2 - 4ac = 1; the paper derives the same conclusion without the condition on p from its Theorem 2.3.

theorem_1_3: Duke's theorem as the paper states it: as d tends to infinity through the positive fundamental discriminants, the probability measure on the union of the closed geodesics attached to d converges to Liouville measure on the unit tangent bundle of the modular surface; the paper reproves it ergodically.

theorem_2_3: The paper's main formulation of Duke's theorem: as d tends to infinity through the non-square discriminants, the normalized measures on the union of the periodic diagonal orbits attached to d converge weak-* to the Haar probability measure; the paper derives from it Skubenko's equidistribution on the hyperboloid with no splitting condition.

theorem_4_2: The paper's finitary form of the uniqueness of the measure of maximal entropy: A-invariant measures with vanishing mass above heights delta_i^(-epsilon) and with few pairs of points within distance delta_i converge to the SL_2(R)-invariant measure.

theorem_5_1: The paper's ergodic form of the statement that high entropy inhibits escape of mass: for M at least some M_0, every invariant probability measure has entropy at most 1 + log log M / log M - mu(X_{>=M})/2, so weak-* limits of measures of entropy at least c keep mass at least 2c - 1.

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Einsiedler, Manfred and Lindenstrauss, Elon and Michel, Philippe and Venkatesh, Akshay, The distribution of closed geodesics on the modular surface, and Duke's theorem. Enseign. Math. (2) 58 (2012), 249--313. DOI: 10.4171/LEM/58-3-2.

The paper reproves Duke's equidistribution theorem for positive discriminants by ergodic means, removing the congruence condition on the discriminant that Linnik and Skubenko had needed. After recalling Linnik's theorem for negative discriminants (Theorem 1.1) and Skubenko's positive-discriminant analog under the hypothesis (d/p) = 1 (Theorem 1.2), the authors state Duke's theorem in the form that the packet G_d of closed geodesics of positive fundamental discriminant d equidistributes on the unit tangent bundle T^1(Y_0(1)) with respect to Liouville measure mu_L (Theorem 1.3), and prove it in the positive-discriminant case. The method replaces Duke's harmonic analysis and Iwaniec's bounds by entropy theory: torus orbits attached to the discriminant are shown to have spacing properties (Section 3), a measure-classification and entropy argument identifies any weak-* limit as the measure of maximal entropy (Section 4 and Appendix B), and positivity of the discriminant is used in place of Linnik's congruence condition, with the escape-of-mass issue handled by an estimate on trajectories spending long time high in the cusp (Proposition 4.3, Section 5). Appendix A treats representations of binary quadratic forms by ternary forms (Proposition 3.4). The paper's own formulation is Theorem 2.3, the equidistribution of the closed orbits attached to every non-square discriminant, with no condition (d/p) = 1 and no restriction to fundamental discriminants; on p. 15 it derives from it the equidistribution of the scaled primitive triples |d|^{-1/2} R_disc(d) on the hyperboloid b^2 - 4ac = 1 in the ratio form of Skubenko's theorem.

Source: https://arxiv.org/abs/1109.0413. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1109.0413), every other right reserved.

Bears on.

• #1148: the paper does not consider the problem. The problem's claim page (Chojecki, 2026) records a proof that uses Duke's theorem in a point-counting form that Chojecki's note deduces from Theorem 2.3; the paper's own deduction (p. 15) is the condition-free equidistribution of |d|^{-1/2} R_disc(d) on b^2 - 4ac = 1 as d tends to infinity through the non-square discriminants.

Results. Labels and pages are those of arXiv:1109.0413v1.

• Theorem 1.2 (Skubenko; p. 3): for a fixed prime p > 2, the scaled primitive forms |d|^{-1/2} R_disc(d) equidistribute on the hyperboloid b^2 - 4ac = 1 as d tends to infinity through the positive discriminants with (d/p) = 1; the paper derives the conclusion without the condition on p. Theorem 1.1 (Linnik, p. 3) is the negative-discriminant analogue.
• Theorem 1.3 (Duke; pp. 4-5): as d tends to infinity through the positive fundamental discriminants, the closed geodesics G_d equidistribute on T^1(Y_0(1)) with respect to Liouville measure; reproved here ergodically.
• Theorem 2.3 (p. 15): as d tends to infinity through the non-square discriminants, the probability measures mu_d on the unions of periodic torus orbits attached to d converge weak-* to the Haar probability measure on PGL_2(Z)\PGL_2(R).
• Proposition 3.6 (Linnik's basic lemma; p. 17): the mu_d x mu_d measure of the pairs below height H within distance delta is <<_eps H^4 delta^3 d^eps for d^{-1/4} <= delta <= H^{-2}/3.
• Theorem 4.2 (p. 23): A-invariant measures with vanishing mass above height delta_i^{-eps} and few delta_i-close pairs converge to the SL_2(R)-invariant measure.
• Proposition 4.3 (p. 23): for a height M >= 1, N >= 1 and V a subset of [-N, N], the points whose trajectory between times -N and N begins and ends below height M and lies above height M exactly at the times in V are covered by <<_M e^{2N - |V|/2} Bowen N-balls, and only <<_M e^{(2 log log M/log M) N} sets V occur.
• Theorem 5.1 (p. 28): for M >= M_0, every invariant probability measure has h_mu(T) <= 1 + log log M/log M - mu(X_{>=M})/2.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above; the labels and pages cited are those of the arXiv preprint arXiv:1109.0413v1.