Related papers: Generalized Thue-Morse Continued Fractions
To a complex polynomial function $f$ with arbitrary singularities we associate the number of Morse points in a general linear Morsification $f_{t} := f - t\ell$. We produce computable algebraic formulas in terms of invariants of $f$ for the…
We give a new proof of Tietze Theorem on the convergence of infinite semi-regular continued fractions.
We define the Thue-Morse transform T on a class of infinite binary words. It sends the alternating word a_0 = 010101... to the Thue-Morse sequence. We then study its orbit a_m = T^m(a_0) as well as the sequences u_m and v_m giving…
We consider a family $\{\tau_m:m\geq 2\}$ of interval maps introduced by Hei-Chi Chan [5] as generalizations of the Gauss transformation. For the continued fraction expansion arising from $\tau_m$, we solve its Gauss-Kuzmin-type problem by…
We develop the local Morse theory for a class of non-twice continuously differentiable functionals on Hilbert spaces, including a new generalization of the Gromoll-Meyer's splitting theorem and a weaker Marino-Prodi perturbation type…
We develop the geometry of Hurwitz continued fractions, a major tool in understanding the approximation properties of complex numbers by ratios of Gaussian integers. Based on a thorough study of the geometric properties of Hurwitz continued…
We investigate a collection of orthonormal functions that encodes information about the continued fraction expansion of real numbers. When suitably ordered these functions form a complete system of martingale differences and are a special…
We reformulate several known results about continued fractions in combinatorial terms. Among them the theorem of Conway and Coxeter and that of Series, both relating continued fractions and triangulations. More general polygon dissections…
We derive closed-form expressions for several new classes of Hurwitzian- and Tasoevian continued fractions, including $[0;\overline{p-1,1,u(a+2nb)-1,p-1,1,v(a+(2n+1)b)-1 }\,\,]_{n=0}^\infty$, $[0; \overline{c + d m^{n}}]_{n=1}^{\infty}$ and…
Infinite products associated with the $\pm 1$ Thue-Morse sequence whose value is rational or algebraic irrational have been studied by several authors. In this short note we prove three new infinite product identities involving ${\pi}$,…
We give continued fraction algorithms for a particular class of Fuchsian triangle groups. In particular, we give an explicit form of each such group that is a subgroup of the Hilbert modular group of its trace field and provide an interval…
It is shown that for sums of functionals of digits in continued fraction expansion the Kolmogorov-Feller weak laws of large numbers and the Khinchine-L\'evy-Feller-Raikov characterization of the domain of attraction of the normal law hold.
The metrical theory of the product of consecutive partial quotients is associated with the uniform Diophantine approximation, specifically to the improvements to Dirichlet's theorem. Achieving some variant forms of metrical theory in…
Let x be a quadratic irrational and let P be the set of prime numbers. We show the existence of an infinite subset S of P such that the statistics of the period of the continued fraction expansions along the sequence {px: p\in S} approach…
In this work, we study a continued fractions theory for the topological completion of the field of Puiseux series. As usual, we prove that any element in the completion can be developed as a unique continued fractions, whose coefficients…
We study a natural extension to complex numbers of the standard continued fractions. The basic algorithm is due to Lagrange and Gauss, though it seems to have gone mostly unnoticed as a way to create continued fractions. The new…
We extend the study of \emph{melonic} quartic tensor models to models with arbitrary quartic interactions. This extension requires a new version of the loop vertex expansion using several species of intermediate fields and iterated…
We prove a continued fraction expansion for the reciprocal of a certain $q$-series. All the specialists in the world are asked whether it is new or not.
We detail the continued fraction expansion of the square root of the general monic quartic polynomial, noting that each line of the expansion corresponds to addition of the divisor at infinity. We analyse the data yielded by the general…
We consider a family of continued fraction expansions of any number in the unit closed interval $[0,1]$ whose digits are differences of consecutive non-positive integer powers of an integer $m \geq 2$. For this expansion, we apply the…