We study the multifractal properties of the uniform approximation exponent and asymptotic approximation exponent in continued fractions. As a corollary, %given a nonnegative reals $\hatν,$ we calculate the Hausdorff dimension of the uniform Diophantine set $$\mathcal{U}(y,\hatν)=\Big\{x\in[0,1)\colon \forall N\gg1, \exists~ n\in[1,N], \text{ such that } |T^{n}(x)-y|<|I_{N}(y)|^{\hatν}\Big\}$$ for algebraic irrational points $y\in[0,1)$. These results contribute to the study of the uniform Diophantine approximation, and apply to investigating the multifractal properties of run-length function in continued fractions.
We provide several results on the diophantine properties of continued fractions on the Heisenberg group, many of which are analogous to classical results for real continued fractions. In particular, we show an analog of Khinchin's theorem and that convergents are also best approximants up to a constant factor of the denominator.
In this part we show that the diophantine equation $X^5+Y^5=\varepsilon^5(1-X^5Y^5)$, where $\varepsilon=\frac{-1+\sqrt{5}}{2}$, has solutions in specific abelian extensions of quadratic fields $K=\mathbb{Q}(\sqrt{-d})$ in which $-d \equiv \pm 1$ (mod $5$). The coordinates of these solutions are values of the Rogers-Ramanujan continued fraction $r(τ)$, and are shown to be periodic points of an algebraic function.
In this paper we study the three-dimensional analogue of the relation between the irrationality exponent of a real number and the growth of its regular continued fraction partial quotients. As a multidimensional generalisation of continued fractions, we consider Klein polyhedra.
We establish arithmetical properties and provide essential bounds for bi-sequences of approximation coefficients associated with the natural extension of maps, leading to continued fraction-like expansions. These maps are realized as the fractional part of M$\operatorname{\ddot{o}}$bius transformations which carry the end points of the unit interval to zero and infinity, extending the classical regular and backwards continued fractions expansions.
While Roth's theorem states that the irrationality measure of all the irrational algebraic numbers is 2, and the same holds true over function fields in characteristic zero, some counter-examples were found over function fields in positive characteristic. This was put forward first by Mahler in 1949, in his fundamental paper on Diophantine approximation \cite{M}. It seems that, except for particular elements, as power series with bounded partial quotients, Roth's theorem holds. Until now, only one element, with unbounded partial quotients, discovered by Mills and Robbins \cite{MR} in 1986, has…
We show that the growth rate of denominator $Q_n$ of the $n$-th convergent of negative expansion of $x$ and the rate of approximation: $$ \frac{\log{n}}{n}\log{\left|x-\frac{P_n}{Q_n}\right|}\rightarrow -\frac{π^2}{3} \quad \text{in measure.} $$ for a.e. $x$. In the course of the proof, we reprove known inspiring results that arithmetic mean of digits of negative continued fraction converges to 3 in measure, although the limit inferior is 2, and the limit superior is infinite almost everywhere.
Gallagher's theorem is a sharpening and extension of the Littlewood conjecture that holds for almost all tuples of real numbers. We provide a fibre refinement, solving a problem posed by Beresnevich, Haynes and Velani in 2015. Hitherto, this was only known on the plane, as previous approaches relied heavily on the theory of continued fractions. Using reduced successive minima in lieu of continued fractions, we develop the structural theory of Bohr sets of arbitrary rank, in the context of diophantine approximation. In addition, we generalise the theory and result to the inhomogeneous setting. …
Our aim is to find a complex continued fraction algorithm finding all the best Diophantine approximations to a complex number. Using the sequence of minimal vectors in a two dimensional lattice over Gaussian integers, we obtain an algorithm defined on a submanifold of the space of unimodular two dimensional Gauss lattices. This submanifold is transverse to the diagonal flow. Thanks to the correspondence between minimal vectors and best Diophantine approximations, the algorithm finds all the best approximations to a complex number. A byproduct of the algorithm is the best constant for the compl…
In 2004, J.C. Tong found bounds for the approximation quality of a regular continued fraction convergent of a rational number, expressed in bounds for both the previous and next approximation. We sharpen his results with a geometric method and give both sharp upper and lower bounds. We also calculate the asymptotic frequency that these bounds occur.
We study quadratic approximations for two families of hyperquadratic continued fractions in the field of Laurent series over a finite field. As the first application, we give the answer to a question of the second author concerning Diophantine exponents for algebraic Laurent series. As the second application, we determine the degrees of these families in particular case.
We show that Y. Cheung's general $Z$-continued fractions can be adapted to give approximation by saddle connection vectors for any compact translation surface. That is, we show the finiteness of his Minkowski constant for any compact translation surface. Furthermore, we show that for a Veech surface in standard form, each component of any saddle connection vector dominates its conjugates. The saddle connection continued fractions then allow one to recognize certain transcendental directions by their developments.
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 map that is piecewise linear fractional, given in terms of group elements. Using natural extensions, we find an ergodic invariant measure for the interval map. We also study diophantine properties of approximation in terms of the continued fractions; and furthermore show that these continued fractions are appropriate to obtain transcendence results.
A {\it two-dimensional continued fraction expansion} is a map $μ$ assigning to every $x \in\mathbb R^2\setminus\mathbb Q^2$ a sequence $μ(x)=T_0,T_1,\dots$ of triangles $T_n$ with vertices $x_{ni}=(p_{ni}/d_{ni},q_{ni}/d_{ni})\in\mathbb Q^2, d_{ni}>0, p_{ni}, q_{ni}, d_{ni}\in \mathbb Z,$ $i=1,2,3$, such that \begin{eqnarray*} \det \left(\begin{matrix} p_{n1}& q_{n1} &d_{n1}\\ p_{n2}& q_{n2} &d_{n2}\\ p_{n3}& q_{n3} &d_{n3} \end{matrix} \right) = \pm 1\,\,\, \,\,\,\mbox{and}\,\,\,\,\,\, \bigcap_n T_n = \{x\}. \end{eqnarray*} We construct a two-dimensional continued fraction expansion $μ^*$ suc…
We exploit dynamical properties of diagonal actions to derive results in Diophantine approximations. In particular, we prove that the continued fraction expansion of almost any point on the middle third Cantor set (with respect to the natural measure) contains all finite patterns (hence is well approximable). Similarly, we show that for a variety of fractals in [0,1]^2, possessing some symmetry, almost any point is not Dirichlet improvable (hence is well approximable) and has property C (after Cassels). We then settle by similar methods a conjecture of M. Boshernitzan saying that there are no …
We study the Diophantine properties of a new class of transcendental real numbers which contains, among others, Roy's extremal numbers, Bugeaud-Laurent Sturmian continued fractions, and more generally the class of Sturmian type numbers. We compute, for each real number $ξ$ of this set, several exponents of Diophantine approximation to the pair $(ξ,ξ^2)$, together with $ω_2^*(ξ)$ and $\widehatω_2^*(ξ)$, the so-called ordinary and uniform exponent of approximation to $ξ$ by algebraic numbers of degree $\leq 2$. As an application, we get new information on the set of values taken by $\widehatω_2^…
There are abundant results on Diophantine approximation over fields of positive characteristic (see the survey papers [13, 25]), but there is very little information about simultaneous approximation. In this paper, we develop a technique of \geometry of numbers" in positive characteristic, so that we may generalize some of the classical results on simultaneous approximation to the case of function fields. More precisely, we approximate a finite set of Laurent series by rational functions with a common denominator. In particular, the lower bound results we obtain may be regarded as a high dimen…
We extend two results about the ordinary continued fraction expansion to best simultaneous Diophantine approximations of vectors or matrices. The first is Levy-Khintchin Theorem about the almost sure growth rate of the denominators of the convergents. The second is a Theorem of Bosma, Hendrik and Wiedijk about the almost sure limit distribution of the sequence of products $q_n d(q_nθ, Z)$ where the $q_n$'s are the denominators of the convergents associated with the real number $θ$ by the ordinary continued fraction algorithm. Beside these two main results, we show that when $d\ge2$, for almost…
We generalize the construction of Roy's Fibonacci type numbers to the case of a Sturmian recurrence and we determine the classical exponents of approximation $ω_2(ξ)$, $\widehatω_2(ξ)$, $λ_2(ξ)$, $\widehatλ_2(ξ)$ associated with these real numbers. This also extends similar results established by Bugeaud and Laurent in the case of Sturmian continued fractions. More generally we provide an almost complete description of the combined graph of parametric successive minima functions defined by Schmidt and Summerer in dimension two for such Sturmian type numbers. As a side result we obtain new info…
For $K$ a cubic field with only one real embedding and $α,β\in K$, we show how to construct an increasing sequence $\{m_n\}$ of positive integers and a subsequence $\{ψ_n\}$ such that (for some constructible constants $C_1,C_2>0$) $\max\{\|m_nα\|,\|m_nβ\|\}<\frac{C_1}{m_n^{1/2}}$ and $\|ψ_nα\|<\frac{C_2}{ψ_n^{1/2}\log ψ_n}$ for all $n$. As a consequence, we have $ψ_n\|ψ_nα\|\|ψ_nβ\|<\frac{C_1 C_2}{\log ψ_n}$, thus giving an effective proof of Littlewood's conjecture for the pair $(α,β)$. Our proofs are elementary and use only standard results from algebraic number theory and the theory of cont…
Zero-one laws are a central topic in metric Diophantine approximation. A classical example of such laws is the Borel-Bernstein theorem. In this note, we prove a complex analogue of the Borel-Bernstein theorem for complex Hurwitz continued fractions. As a corollary, we obtain a complex version of Khinchin's theorem on Diophantine approximation.
The continuous version of a fundamental result of Khinchin says that a half-infinite torus line in the unit square $[0,1]^2$ exhibits superdensity, which is a best form of time-quantitative density, if and only if the slope of the geodesic is a badly approximable number. In this paper, we give a proof of the extension of this result of Khinchin to the case when the unit torus $[0,1]^2$ is replaced by a finite polysquare surface, or square tiled surface. The argument is based on diophantine approximation and continued fractions, traditional tools in number theory. In particular, we use the famo…
We investigate the distribution of $pθ$ modulo 1, where $θ$ is a complex number which is not contained in $\mathbb{Q}(i)$, and $p$ runs over the Gaussian primes.
All the periodic points of a certain algebraic function related to the Rogers-Ramanujan continued fraction $r(τ)$ are determined. They turn out to be $0, \frac{-1 \pm \sqrt{5}}{2}$, and the conjugates over $\mathbb{Q}$ of the values $r(w_d/5)$, where $w_d$ is one of a specific set of algebraic integers, divisible by the square of a prime divisor of 5, in the field $K_d=\mathbb{Q}(\sqrt{-d})$, as $-d$ ranges over all negative quadratic discriminants for which $\left(\frac{-d}{5}\right) = +1$. This yields new insights on class numbers of orders in the fields $K_d$. Conjecture 1 of Part I is prov…
We give an algorithm that finds a sequence of approximations with Dirichlet coefficients bounded by a constant only depending on the dimension. The algorithm uses the LLL-algorithm for lattice basis reduction. We present a version of the algorithm that runs in polynomial time of the input.
We initiate the study of an intrinsic notion of Diophantine approximation on a rational Carnot group $G$. If $G$ has Hausdorff dimension $Q$, we show that its Diophantine exponent is equal to $(Q+1)/Q$, generalizing the case $G=\mathbb R^n$. We furthermore obtain a precise asymptotic on the count of rational approximations. We then focus on the case of the Heisenberg group $\mathbb H^n$, distinguishing between two notions of Diophantine approximation by rational points in $\mathbb H^n$: Carnot Diophantine approximation and Siegel Diophantine approximation. After computing the Siegel Diophantin…
A Trott number is a number $x\in(0,1)$ whose continued fraction expansion is equal to its base $b$ expansion for a given base $b$, in the following sense: If $x=[0;a_1,a_2,\dots]$, then $x=(0.\hat{a}_1\hat{a}_2\dots)_b$, where $\hat{a}_i$ is the string of digits resulting from writing $a_i$ in base $b$. In this paper we characterize the set of bases for which Trott numbers exist, and show that for these bases, the set $T_b$ of Trott numbers is a complete $G_δ$ set. We prove moreover that the union $T:=\bigcup_{b\geq 2} T_b$ is nowhere dense and has Hausdorff dimension less than one. Finally, w…
The paper deals with best one--sided (lower or upper) Diophantine approximations of the $\ell$-th kind ($\ell\in\mathbb{N}$). We use the ordinary continued fraction expansions to formulate explicit criteria for a fraction $\frac{p}{q}\in\mathbb{Q}$ to be a best lower or upper Diophantine approximation of the $\ell$-th kind to a given $α\in\mathbb{R}$. The sets of best lower and upper approximations are examined in terms of their cardinalities and metric properties. Applying our results in spectral analysis, we obtain an explanation for the rarity of so-called Bethe--Sommerfeld quantum graphs.
We study the problem of Diophantine approximation on lines in $\mathbb{C}^2$ with numerators and denominators restricted to Gaussian primes. To this end, we develop analogs of well-known results on small fractional parts of $pγ$, $p$ running over the primes and $γ$ being a fixed irrational, for Gaussian primes.
Let $\mathscr{L}(S^1)$ be the Lagrange spectrum arising from intrinsic Diophantine approximation on the unit circle $S^1$ by its rational points. We give a complete description of the structure of $\mathscr{L}(S^1)$ below its smallest accumulation point. To this end, we use digit expansions of points on $S^1$, which were originally introduced by Romik in 2008 as an analogue of simple continued fraction of a real number. We prove that the smallest accumulation point of $\mathscr{L}(S^1)$ is 2. Also we characterize the points on $S^1$ whose Lagrange numbers are less than 2 in terms of Romik's di…
This thesis is about the study of Diophantine equations involving binary recurrent sequences with arithmetic functions. Various Diophantine problems are investigated and new results are found out of this study. Firstly, we study several questions concerning the intersection between two classes of non-degenerate binary recurrence sequences and provide, whenever possible, effective bounds on the largest member of this intersection. Our main study concerns Diophantine equations of the form $φ(|au_n |)=|bv_m|,$ where $φ$ is the Euler totient function, $\{u_n\}_{n\geq 0}$ and $\{v_m\}_{m\geq 0}$ ar…
Given for instance a finite volume negatively curved Riemannian manifold $M$, we give a precise relation between the logarithmic growth rates of the excursions into cusps neighborhoods of the strong unstable leaves of negatively recurrent unit vectors of $M$ and their linear divergence rates under the geodesic flow. As an application to non-Archimedian Diophantine approximation in positive characteristic, we relate the growth of the orbits of lattices under one-parameter unipotent subgroups of $\GL_2(\wh K)$ with approximation exponents and continued fraction expansions of elements of the fiel…
We generalise M. M. Skriganov's notion of weak admissibility for lattices to include standard lattices occurring in Diophantine approximation and algebraic number theory, and we prove estimates for the number of lattice points in sets such as aligned boxes. Our result improves on Skriganov's celebrated counting result if the box is sufficiently distorted, the lattice is not admissible, and, e.g., symplectic or orthogonal. We establish a criterion under which our error term is sharp, and we provide examples in dimensions $2$ and $3$ using continued fractions. We also establish a similar countin…
In this paper we give a detailed measure theoretical analysis of what we call sum-level sets for regular continued fraction expansions. The first main result is to settle a recent conjecture of Fiala and Kleban, which asserts that the Lebesgue measure of these level sets decays to zero, for the level tending to infinity. The second and third main result then give precise asymptotic estimates for this decay. The proofs of these results are based on recent progress in infinite ergodic theory, and in particular, they give non-trivial applications of this theory to number theory. The paper closes …
In this paper, we study Diophantine exponents $w_n$ and $w_n ^{*}$ for Laurent series over a finite field. Especially, we deal with the case $n=2$, that is, quadratic approximation. We first show that the range of the function $w_2-w_2 ^{*}$ is exactly the closed interval $[0,1]$. Next, we estimate an upper bound of the exponent $w_2$ of continued fractions with low complexity partial quotients.
Markov numbers are integers that appear in the solution triples of the Diophantine equation, $x^2+y^2+z^2=3xyz$, called the Markov equation. A classical topic in number theory, these numbers are related to many areas of mathematics such as combinatorics, hyperbolic geometry, approximation theory and cluster algebras. There is a natural map from the rational numbers between zero and one to the Markov numbers. In this paper, we prove two conjectures seen in Martin Aigner's book, Markov's theorem and 100 years of the uniqueness conjecture, that determine an ordering on subsets of the Markov numbe…
