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### A note on counting cuspidal excursions.

Annales Academiae Scientiarum Fennicae. Series A I. Mathematica

### Asymptotic growth of Markoff-Hurwitz numbers

Compositio Mathematica

### Closed Curves and Geodesics with Two Self-Intersections on the Punctured Torus.

Monatshefte für Mathematik

Acta Arithmetica

### Correction to : “Linear fractional transformations of continued fractions with bounded partial quotients”

Journal de théorie des nombres de Bordeaux

We fill a gap in the proof of a theorem of our paper cited in the title.

### Galleries in Poincaré half-spaces and diophantine approximation.

Beiträge zur Algebra und Geometrie

### Generalized Lyndon words. (Mots de Lyndon généralisés.)

Séminaire Lotharingien de Combinatoire [electronic only]

### Linear fractional transformations of continued fractions with bounded partial quotients

Journal de théorie des nombres de Bordeaux

Let $\theta$ be a real number with continued fraction expansion$\theta =\left[{a}_{0},{a}_{1},{a}_{2},\cdots \right],$and let$M=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$be a matrix with integer entries and nonzero determinant. If $\theta$ has bounded partial quotients, then $\frac{a\theta +b}{c\theta +d}=\left[{a}_{0}^{*},{a}_{1}^{*},{a}_{2}^{*},\cdots \right]$ also has bounded partial quotients. More precisely, if ${a}_{j}\le K$ for all sufficiently large $j$, then ${a}_{j}^{*}\le |det\left(M\right)|\left(K+2\right)$ for all sufficiently large $j$. We also give a weaker bound valid for all ${a}_{j}^{*}$ with $j\ge 1$. The proofs use the homogeneous Diophantine approximation constant ${L}_{\infty }\left(\theta \right)={lim sup}_{q\to \infty }{\left(q∥{q}^{\theta }∥\right)}^{-1}$. We show that$\frac{1}{\left|det\left(M\right)\right|}{L}_{\infty }\left(\theta \right)\le {L}_{\infty }\left(\frac{a\theta +b}{c\theta +d}\right)\le \left|det\left(M\right)\right|{L}_{\infty }\left(\theta \right).$

### Location of approximations of a Markoff theorem.

International Journal of Mathematics and Mathematical Sciences

### Markoff numbers and ambiguous classes

Journal de Théorie des Nombres de Bordeaux

The Markoff conjecture states that given a positive integer $c$, there is at most one triple $\left(a,b,c\right)$ of positive integers with $a\le b\le c$ that satisfies the equation ${a}^{2}+{b}^{2}+{c}^{2}=3abc$. The conjecture is known to be true when $c$ is a prime power or two times a prime power. We present an elementary proof of this result. We also show that if in the class group of forms of discriminant $d=9{c}^{2}-4$, every ambiguous form in the principal genus corresponds to a divisor of $3c-2$, then the conjecture is true. As a result, we obtain criteria in terms of...

### Matrix theoretic interpretation of the classical Markoff theory. (L'interprétation matricielle de la théorie de Markoff classique.)

International Journal of Mathematics and Mathematical Sciences

Acta Arithmetica

### On Perrine's generalized Markoff equation.

Aequationes mathematicae

### On Perrine's generalized Markoff equations. (Summary).

Aequationes mathematicae

### On the closedness of approximation spectra

Journal de Théorie des Nombres de Bordeaux

Generalizing Cusick’s theorem on the closedness of the classical Lagrange spectrum for the approximation of real numbers by rational ones, we prove that various approximation spectra are closed, using penetration properties of the geodesic flow in cusp neighbourhoods in negatively curved manifolds and a result of Maucourant [Mau].

### On the Gaps of the Markoff Spectrum.

Monatshefte für Mathematik

### On the greatest prime factor of Markov pairs

Rendiconti del Seminario Matematico della Università di Padova

### On the inhomogeneous Hall's ray of period-one quadratics.

Experimental Mathematics

### Sur des équations diophantiennes généralisant celle de Markoff

Annales de la Faculté des sciences de Toulouse : Mathématiques

### Systoles of arithmetic surfaces and the Markoff spectrum.

Mathematische Annalen

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