This problem was asked to K. Mahler by one of his Japanese colleagues, a Z-number is a positive real number $x$ such that the fractional parts of $x(\frac{3}{2})^n $ are less than $\frac{1}{2}$ for all integers $n$ such that $n \ge 0$. Kurt Mahler conjectured in 1968 that there are no Z-numbers. In this paper, we show that there are no Z-numbers in $\mathbb{Z}^{+} = \{1,2,3,...\}$.
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…
Let $q$ a prime power and ${\mathbb F}_q$ the finite field of $q$ elements. We study the analogues of Mahler's and Koksma's classifications of complex numbers for power series in ${\mathbb F}_q((T^{-1}))$. Among other results, we establish that both classifications coincide, thereby answering a question of Ooto.
We provide a general result for the algebraic independence of Mahler functions by a new method based on asymptotic analysis. As a consequence of our method, these results hold not only over $\mathbb{C}(z)$, but also over $\mathbb{C}(z)(\mathcal{M})$, where $\mathcal{M}$ is the set of all meromorphic functions. Several examples and corollaries are given, with special attention to nonnegative regular functions.
In this paper, we shall prove that any subset of $\overline{\mathbb Q}\cap B(0,1)$, which is closed under complex conjugation and which contains the element $0$, is the exceptional set of uncountably many transcendental functions, analytic in the unit ball, with integer coefficients. This solves a strong version of an old question proposed by K. Mahler (1976).
For an algebraic number $α$, the metric Mahler measure $m_1(α)$ was first studied by Dubickas and Smyth in 2001 and was later generalized to the $t$-metric Mahler measure $m_t(α)$ by the author in 2010. The definition of $m_t(α)$ involves taking an infimum over a certain collection $N$-tuples of points in $\overline{\mathbb Q}$, and from previous work of Jankauskas and the author, the infimum in the definition of $m_t(α)$ is attained by rational points when $α\in \mathbb Q$. As a consequence of our main theorem in this article, we obtain an analog of this result when $\mathbb Q$ is replaced wi…
We consider Mahler functions $f(z)$ which solve the functional equation $f(z) = \frac{A(z)}{B(z)} f(z^d)$ where $\frac{A(z)}{B(z)}\in \mathbb{Q}(z)$ and $d\ge 2$ is integer. We prove that for any integer $b$ with $|b|\ge 2$ either $f(b)$ is rational or its irrationality exponent is rational. We also compute the exact value of the irrationality exponent for $f(b)$ as soon as the continued fraction for the corresponding Mahler function is known. This improves the result of Bugeaud, Han, Wei and Yao where only an upper bound for the irrationality exponent was provided.
In 1902, P. Stäckel proved the existence of a transcendental function $f(z)$, analytic in a neighbourhood of the origin, and with the property that both $f(z)$ and its inverse function assume, in this neighbourhood, algebraic values at all algebraic points. Based on this result, in 1976, K. Mahler raised the question of the existence of such functions which are analytic in $\mathbb{C}$. Recently, the authors answered positively this question. In this paper, we prove a much stronger version of this result by considering other subsets of $\mathbb{C}$.
We prove a new lower bound for the Mahler measure of a polynomial in one and in several variables that depends on the complex coefficients, and the number of monomials. In one variable our result generalizes a classical inequality of Mahler. In $M$ variables our result depends on $\mathbb{Z}^M$ as an ordered group, and in general our lower bound depends on the choice of ordering.
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 t…
Given an elliptic curve $E$ defined over $\mathbb{Q}$ which has potential complex multiplication by the ring of integers $\mathcal{O}_K$ of an imaginary quadratic field $K$ we construct a polynomial $P_E \in \mathbb{Z}[x,y]$ which is a planar model of $E$ and such that the Mahler measure $m(P_E) \in \mathbb{R}$ is related to the special value of the $L$-function $L(E,s)$ at $s = 2$.
Let $p$ be a prime number. We develop a theory of $p$-adic Mahler measure of polynomials and apply it to the study of $\mathbb{Z}$-covers of rational homology 3-spheres branched over links. We obtain a $p$-adic analogue of the asymptotic formula of the torsion homology growth and a balance formula among the leading coefficient of the Alexander polynomial, the $p$-adic entropy, and the Iwasawa $μ_p$-invariant. We also apply the purely $p$-adic theory of Besser--Deninger to $\mathbb{Z}$-covers of links. In addition, we study the entropies of profinite cyclic covers of links. We examine various e…
This note deals with some effective results in Mahler's method. In a recent work, we used a theorem of Philippon to show that given a Mahler function $f(z)$ in ${\bf k}\{z\}$, where ${\bf k}$ denotes a number field, and an algebraic number $α$ in the domain of holomorphy of $f$, the number $f(α)$ is either transcendental or belongs to ${\bf k}(α)$. We describe here an effective procedure to decide if such a number is transcendental or not. More generally, given several Mahler functions $f_1(z),\cdots,f_r(z)$ and an algebraic number $α$ in the domain of holomorphy of these functions, we show ho…
Let $F \in \mathbb Z[x, y]$ be an irreducible binary form of degree $d \geq 7$ and content one. Let $α$ be a root of $F(x, 1)$ and assume that the field extension $\mathbb Q(α)/\mathbb Q$ is Galois. We prove that, for every sufficiently large prime power $p^k$, the number of solutions to the Diophantine equation of Thue type $$ |F(x, y)| = tp^k $$ in integers $(x, y, t)$ such that $\gcd(x, y) = 1$ and $1 \leq t \leq (p^k)^λ$ does not exceed $24$. Here $λ= λ(d)$ is a certain positive, monotonously increasing function that approaches one as $d$ tends to infinity. We also prove that, for every su…
For $p \in \mathbb{Q}_+ \smallsetminus \{ 1 \}$ a positive rational number different from one, we say that the Puisseux series $f \in \mathbb{C}((t))^\text{alg}$ is $p$-Mahler of non-exceptional polynomial type if there is a polynomial $P \in \mathbb{C}(t)^\text{alg}[X]$ of degree at least two which is not conjugate to either a monomial or to plus or minus a Chebyshev polynomial for which the equation $f(t^p) = P(f(t))$ holds. We show that if $p$ and $q$ are multiplicatively independent and $f$ and $g$ are $p$-Mahler and $q$-Mahler, respectively, of non-exceptional polynomial type, then $f$ an…
If $α$ is a non-zero algebraic number, we let $m(α)$ denote the Mahler measure of the minimal polynomial of $α$ over $\mathbb Z$. A series of articles by Dubickas and Smyth, and later by the author, develop a modified version of the Mahler measure called the $t$-metric Mahler measure, denoted $m_t(α)$. For fixed $α\in \bar{\mathbb Q}$, the map $t\mapsto m_t(α)$ is continuous, and moreover, is infinitely differentiable at all but finitely many points, called {\it exceptional points} for $α$. It remains open to determine whether there is a sequence of elements $α_n\in \bar{\mathbb Q}$ such that …
For an algebraic number $α$ we denote by $M(α)$ the Mahler measure of $α$. As $M(α)$ is again an algebraic number (indeed, an algebraic integer), $M(\cdot)$ is a self-map on $\overline{\mathbb{Q}}$, and therefore defines a dynamical system. The \emph{orbit size} of $α$, denoted $\# \mathcal{O}_M(α)$, is the cardinality of the forward orbit of $α$ under $M$. We prove that for every degree at least 3 and every non-unit norm, there exist algebraic numbers of every orbit size. We then prove that for algebraic units of degree 4, the orbit size must be 1, 2, or infinity. We also show that there exis…
This is the second paper in a series of two in which a global algebraic number theory of the reals is formulated with the purpose of providing a unified setting for algebraic and transcendental number theory. In this paper, to any real number $θ$ we associate its polynomial diophantine approximation ring: a tri-filtered subring of a nonstandard model of the ring $\mathbb{Z}[X]$. We characterize the filtration structure of the polynomial diophantine approximation ring according to the Mahler class and the Mahler type of $θ$. The arithmetic of polynomial diophantine approximation groups is intro…
We provide an ergodicity criterion for uniformly differentiable modulo $p$ functions on ${\mathbb Z}_p$ in regard to the minimal level of the reduced functions by showing that ergodic conditions are explicitly found in terms of the coefficients of Mahler or van der Put for each prime $p.$ To this end, it is essential to give an alternative, natural proof of Memi$\acute{c}$'s result regarding Mahler's coefficients estimation for uniformly differentiable modulo $p$ functions on ${\mathbb Z}_p.$
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…
For every $P \in \mathbb{Z}[x_1^{\pm 1}, \ldots, x_d^{\pm 1}] \setminus \{0\}$, and every $\varepsilon > 0$, we prove that there are a computable function $M = M(d,\varepsilon,°{P},h(P)) < \infty$ and a finite union $Z = Z(d,\varepsilon,°{P},h(P))$ of proper torsion cosets $\boldsymbolμ T \subsetneq \mathbb{G}_m^d$ such that, for every $N \in \mathbb{N}$, $Z$ contains all but at most $M$ of the torsion points $\boldsymbolζ \in μ_N^d$ satisfying $|P(\boldsymbolζ)| < e^{-\varepsilon ϕ(N)}$. This extends a well known structural theorem from torsion points lying exactly on a variety to torsion poi…
We show that the minimal positive logarithmic Lind-Mahler measure for a group of the form $G=\mathbb Z_2^r\times\mathbb Z_4^s$ with $|G|\geq 4$ is $\frac{1}{|G|} \log (|G|-1).$ We also show that for $G=\mathbb Z_2 \times \mathbb Z_{2^n}$ with $n\geq 3$ this value is $\frac{1}{|G|} \log 9.$ Previously the minimal measure was only known for $2$-groups of the form $\mathbb Z_2^k$ or $\mathbb Z_{2^k}.$
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…
This work addresses problems on simultaneous Diophantine approximation on fractals, motivated by a long standing problem of Mahler regarding Cantor's middle $1/3$ set. We obtain the first instances where a complete analogue of Khintchine's Theorem holds for fractal measures. Our results apply to fractals which are self-similar by a system of rational similarities of $\mathbb{R}^d$ (for any $d\geq 1$) and have sufficiently small Hausdorff co-dimension. A concrete example of such measures in the context of Mahler's problem is the Hausdorff measure on the "middle $1/5$ Cantor set"; i.e. the set o…
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…
The reduction modulo $p$ of a family of lacunary integer polynomials, associated with the dynamical zeta function $ζ_β(z)$ of the $β$-shift, for $β> 1$ close to one, is investigated. We briefly recall how this family is correlated to the problem of Lehmer. A variety of questions is raised about their numbers of zeroes in $\mathbb{F}_p$ and their factorizations, via Kronecker's Average Value Theorem (viewed as an analog of classical Theorems of Uniform Distribution Theory). These questions are partially answered using results of Schinzel, revisited by Sawin, Shusterman and Stoll, and density th…
For each algebraic number $α$ and each positive real number $t$, the $t$-metric Mahler measure $m_t(α)$ creates an extremal problem whose solution varies depending on the value of $t$. The second author studied the points $t$ at which the solution changes, called {\it exceptional points for $α$}. Although each algebraic number has only finitely many exceptional points, it is conjectured that, for every $N \in \mathbb N$, there exists a number having at least $N$ exceptional points. In this article, we describe a polynomial time algorithm for establishing the existence of numbers with at least …
