The Mathematical-Function Computation Handbook

Conjecturally, the only real algebraic numbers with bounded partial quotients in their regular continued fraction expansion are rationals and quadratic irrationals. We show that the corresponding statement is not true for complex algebraic numbers in a very strong sense, by constructing, for every even degree that have bounded complex partial quotients in their Hurwitz continued fraction expansion. The Hurwitz expansion is the complex generalization of the nearest integer continued fraction for real numbers. In the case of real numbers the boundedness of regular and nearest integer partial quotients is equivalent.

/pdf/complex-numbers-with-bounded-partial-quotients-3u60tg01ju.pdf

Complex numbers with bounded partial quotients

Let L be the transfer operator associated with the Gauss' continued fraction map, known also as the Gauss-Kuzmin-Wirsing operator, acting on the Banach space. In this work we prove an asymptotic formula for the eigenvalues of L. This settles, in a stronger form, the conjectures of D. Mayer and G. Roepstorff (1988), A.J. MacLeod (1992), Ph. Flajolet and B. Vallee (1995), also supported by several other authors. Further, we find an exact series for the eigenvalues, which also gives the canonical decomposition of trace formulas due to D. Mayer (1976) and K.I. Babenko (1978). This crystallizes the contribution of each individual eigenvalue in the trace formulas.

Transfer operator for the Gauss' continued fraction map. I. Structure of the eigenvalues and trace formulas

The Gauss-Kuzmin statistics for the triangle map (a type of multidimensional continued fraction algorithm) are derived by examining the leading eigenfunction of the triangle map's transfer operator. The technical difficulty is finding the appropriate Banach space of functions. We also show that, by thinking of the triangle map's transfer operator as acting on a one-dimensional family of Hilbert spaces, the transfer can be thought of as a family of nuclear operators of trace class zero.

On Gauss-Kuzmin Statistics and the Transfer Operator for a Multidimensional Continued Fraction Algorithm: the Triangle Map

We provide a refinement of the Poincare inequality on the torus $\mathbb{T}^{d}$
 : there exists a set $\mathcal{B} \subset \mathbb{T} ^{d}$
 of directions such that for every $\alpha \in \mathcal{B}$
 there is a $c_{\alpha } > 0$
 with $$\begin{aligned} \|\nabla f\|_{L^{2}(\mathbb{T}^{d})}^{d-1} \| \langle \nabla f, \alpha \rangle \|_{L^{2}(\mathbb{T}^{d})} \geq c_{\alpha }\|f\| _{L^{2}(\mathbb{T}^{d})}^{d} \quad \mbox{for all}~f\in H^{1}\bigl( \mathbb{T}^{d}\bigr)~ \mbox{with mean 0.} \end{aligned}$$ The derivative $\langle \nabla f, \alpha \rangle $
 does not detect any oscillation in directions orthogonal to $\alpha $
 , however, for certain $\alpha $
 the geodesic flow in direction $\alpha $
 is sufficiently mixing to compensate for that defect. On the two-dimensional torus $\mathbb{T}^{2}$
 the inequality holds for $\alpha = (1, \sqrt{2})$
 but is not true for $\alpha = (1,e)$
 . Similar results should hold at a great level of generality on very general domains.

/pdf/directional-poincare-inequalities-along-mixing-flows-59ld5jz963.pdf

Directional Poincaré inequalities along mixing flows

Complex Continued Fractions

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Complex Continued Fractions

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Citations

The Mathematical-Function Computation Handbook

Complex numbers with bounded partial quotients

Transfer operator for the Gauss' continued fraction map. I. Structure of the eigenvalues and trace formulas

On Gauss-Kuzmin Statistics and the Transfer Operator for a Multidimensional Continued Fraction Algorithm: the Triangle Map

Directional Poincaré inequalities along mixing flows

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