Let $θ$ and $ρ$ be real numbers with $0 \le θ, ρ< 1$ and $θ$ irrational. We show that the Hecke-Mahler series $$ F_{θ, ρ} (z_1, z_2) = \sum_{k_1 \ge 1} \, \sum_{k_2 = 1}^{\lfloor k_1 θ+ ρ\rfloor} \, z_1^{k_1} z_2^{k_2}, $$ where $\lfloor \cdot \rfloor$ denotes the integer part function, takes transcendental values at any algebraic point $(β, α)$ with $0 < |β|, |βα^θ| < 1$. This extends earlier results of Mahler (1929) and Loxton and van der Poorten (1977), who settled the case $ρ=0$. Furthermore, for positive integers $b$ and $a$, with $b \ge 2$ and $a$ congruent to $1$ modulo $b-1$, we give the continued fraction expansion of the number
$$
{(b-1)^2\over b} F_{θ, ρ} \left({1\over b}, {1\over a}\right)+{\lfloor θ+ρ\rfloor(b-1)\over b^2a},
$$
from which we derive a formula giving the irrationality exponent of $F_{θ, ρ} (1/b, 1/a)$.
Here we propose a survey on Mahler's theory for transcendence and algebraic independence focusing on certain applications to the arithmetic of periods of Anderson t-motives.
We present a completely explicit transcendence measure for $e$. This is a continuation and an improvement to the works of Borel, Mahler and Hata on the topic. Furthermore, we also prove a transcendence measure for an arbitrary positive integer power of $e$. The results are based on Hermite-Padé approximations and on careful analysis of common factors in the footsteps of Hata.
We give two tests for transcendence of Mahler functions. For our first, we introduce the notion of the eigenvalue $λ_F$ of a Mahler function $F(z)$, and develop a quick test for the transcendence of $F(z)$ over $\mathbb{C}(z)$, which is determined by the value of the eigenvalue $λ_F$. While our first test is quick and applicable for a large class of functions, our second test, while a bit slower than our first, is universal; it depends on the rank of a certain Hankel matrix determined by the initial coefficients of $F(z)$. We note that these are the first transcendence tests for Mahler functio…
We present short proofs of the transcendence of the Liouville and the Mahler numbers. The first proof is known for a long time, the second proof apparently appeared only in 2002-2003. The proofs are accessible to high-school students.
Here we characterise in a complete and explicit way the relations of algebraic dependence over $\QQ$ of complex values of Hecke-Mahler series taken at algebraic points of the multiplicative group $\GG_m^2(\CC)$. Our result contains previous theorems by Loxton and van der Poorten, Mahler, and Masser.
For a given transcendental number $ξ$ and for any polynomial $P(X)=: λ_0+\cdots+λ_k X^k \in \mathbb{Z}[X]$, we know that $ P(ξ) \neq 0.$ Let $k \geq 1$ and $ω(k, H)$ be the infimum of the numbers $r > 0$ satisfying the estimate $$ \left|λ_0+λ_1 ξ+λ_2 ξ^{2}+ \ldots +λ_kξ^{k}\right| > \frac{1}{H^r}, $$ for all $(λ_0, \ldots ,λ_k)^T \in \mathbb{Z}^{k+1}\setminus\{\overline{0}\}$ with $\max_{1\le i\le k} \{|λ_i|\} \le H$. Any function greater than or equal to $ω(k, H)$ is a {\it transcendence measure of $ξ$}. In this article, we find out a transcendence measure of $ e^{1/n}$ which improves a resul…
Let $f : [0,1)\rightarrow [0,1)$ be a $2$-interval piecewise affine increasing map which is injective but not surjective. Such a map $f$ has a rotation number and can be parametrized by three real numbers. We make fully explicit the dynamics of $f$ thanks to two specific functions $δ$ and $ϕ$ depending on these parameters whose definitions involve Hecke-Mahler series. As an application, we show that the rotation number of $f$ is rational, when the three parameters are algebraic numbers.
Euler gave recipes for converting alternating series of two types, I and II, into equivalent continued fractions, i.e., ones whose convergents equal the partial sums. A condition we prove for irrationality of a continued fraction then allows easy proofs that $e,\sin1$, and the primorial constant are irrational. Our main result is that, if a series of type II is equivalent to a simple continued fraction, then the sum is transcendental and its irrationality measure exceeds $2$. We construct all $\aleph_0^{\aleph_0}=\mathfrak{c}$ such series and recover the transcendence of the Davison--Shallit a…
This is the first part of a work devoted to the study of linear Mahler systems in several variables from the perspective of transcendence and algebraic independence. We prove two main results concerning systems that are regular singular at the origin. Our interest in Mahler's method comes from the possible applications of these results to old problems involving automata theory and which concern the expansion of both natural numbers and real numbers in integer bases. In particular, problems which involve finite automata and base change. Such applications are studied in Part II of this work.
Let $\mathbb{K}$ be a function field of characteristic $p>0$. We recently established the analogue of a theorem of Ku. Nishioka for linear Mahler systems defined over $\mathbb{K}(z)$. This paper is dedicated to proving the following refinement of this theorem. Let $f_{1}(z),\ldots f_{n}(z)$ be $d$-Mahler functions such that $\overline{\mathbb{K}}(z)\left(f_{1}(z),\ldots, f_{n}(z)\right)$ is a regular extension over $\overline{\mathbb{K}}(z)$. Then, every homogeneous algebraic relation over $\overline{\mathbb{K}}$ between their values at a regular algebraic point arises as the specialization of…
{\em The Liouville number}, denoted $l$, is defined by $$l:=0.100101011101101111100...,$$ where the $n$th bit is given by ${1/2}(1+\gl(n))$; here $\gl$ is the Liouville function for the parity of prime divisors of $n$. Presumably the Liouville number is transcendental, though at present, a proof is unattainable. Similarly, define {\em the Gaussian Liouville number} by $$γ:=0.110110011100100111011...$$ where the $n$th bit reflects the parity of the number of rational Gaussian primes dividing $n$, 1 for even and 0 for odd. In this paper, we prove that the Gaussian Liouville number and its relati…
We express the logarithmic Mahler measure of two families of polynomials defining K3 hypersurfaces of generic Picard number 19 in terms of Eisenstein-Kronecker series. For some singular K3 surfaces (i.e of Picard number 20) we write this measure as Hecke L-series of weight 3 with "Gr\" ossencharacters".
This is the second part of a work devoted to the study of linear Mahler systems in several variables from the perspective of transcendence and algebraic independence. From the lifting theorem obtained in the first part, we first derive a general result, showing that Mahler functions in several variables, associated with transformations having multiplicatively dependent spectral radii, take algebraic independent values at algebraic points provided that these points are sufficiently independent. Then, we focus on applications of this result and of the two main results of Part I of this work. Our…
We develop a theory of linear Mahler systems in several variables from the perspective of transcendence and algebraic independence, which also includes the possibility of dealing with several systems associated with sufficiently independent matrix transformations. Our main results go far beyond the existing literature, also surpassing those of two unpublished preprints the authors made available on the arXiv in 2018. The main new feature is that they apply now without any restriction on the matrices defining the corresponding Mahler systems. As a consequence, we settle several problems concern…
Let $F_1=1,F_2=1,\ldots$ be the Fibonacci sequence. Motivated by the identity $\displaystyle\sum_{k=0}^{\infty}\frac{1}{F_{2^k}}=\frac{7-\sqrt{5}}{2}$, Erdös and Graham asked whether $\displaystyle\sum_{k=1}^{\infty}\frac{1}{F_{n_k}}$ is irrational for any sequence of positive integers $n_1,n_2,\ldots$ with $\frac{n_{k+1}}{n_k}\geq c>1$. We resolve the transcendence counterpart of their question: as a special case of our main theorem, we have that $\displaystyle\sum_{k=1}^{\infty}\frac{1}{F_{n_k}}$ is transcendental when $\frac{n_{k+1}}{n_k}\geq c>2$. The bound $c>2$ is best possible thanks to…
Let $0<λ<1$. We consider the one-parameter family of circle $λ$-affine contractions $f_δ:x \in [0,1) \mapsto λx + δ\; {\rm mod}\,1 $, where $0 \le δ<1$. Let $ρ$ be the rotation number of the map $f_δ$. We will give some numerical relations between the values of $λ,δ$ and $ρ$, essentially using Hecke-Mahler series and a tree structure. When both parameters $λ$ and $δ$ are algebraic numbers, we show that $ρ$ is a rational number. Moreover, in the case $λ$ and $δ$ are rational, we give an explicit upper bound for the height of $ρ$ under an assumption on $λ$.
