Boris Adamczewski, Jason P. Bell
Let $K$ be a field of characteristic zero and $k$ and $l$ be two multiplicatively independent positive integers. We prove the following result that was conjectured by Loxton and van der Poorten during the Eighties: a power series $F(z)\in K[[z]]$ satisfies both a $k$- and a $l$-Mahler type functional equation if and only if it is a rational function.
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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…
In 1969, I. Bernstein and S. Gelfand introduced an object, which is now called the zeta Mahler function (ZMF, also zeta Mahler measure) and related to the Mahler measure. Here we discuss a family of ZMFs attached to the Laurent polynomials $k + (x_1 + x_1^{-1}) \cdots (x_r + x_r^{-1})$, where $k$ is real. We give explicit formulae, present examples and establish properties for these ZMFs, such as an RH-type phenomenon. Further, we explore relations with the Mahler measure.
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).
A strange identity of an MF (Mahler function)
本文研究了 Mahler 方程的两个不同解，建立了仅在单位根处定义的解与单位圆盘内解析解之间的奇特恒等关系。
We give an upper bound for the zero order of the difference between a Mahler function and an algebraic function. This complements estimates of Nesterenko, Nishioka, and Töpfer, among others, who considered polynomials evaluated at Mahler functions.
In the theory of continued fractions, Zaremba's conjecture states that there is a positive integer $M$ such that each integer is the denominator of a convergent of an ordinary continued fraction with partial quotients bounded by $M$. In this paper, to each such $M$ we associate a regular sequence---in the sense of Allouche and Shallit---and establish various properties and results concerning the generating function of the regular sequence. In particular, we determine the minimal algebraic relation concerning the generating function and its Mahler iterates.
We consider a generalization of the Mahler measure of a multivariable polynomial $P$ as the integral of $\log^k|P|$ in the unit torus, as opposed to the classical definition with the integral of $\log|P|$. A zeta Mahler measure, involving the integral of $|P|^s$, is also considered. Specific examples are computed, yielding special values of zeta functions, Dirichlet $L$-functions, and polylogarithms.
Repeatedly folding a strip of paper in half and unfolding it in straight angles produces a fractal: the dragon curve. Shallit, van der Poorten and others showed that the sequence of right and left turns relates to a continued fraction that is also a simple infinite series. We construct a Mahler function from two functions of Dilcher and Stolarsky with similar properties. It produces a predictable irregular continued fraction that admits a regular continued fraction and a shape resembling the dragon curve. Furthermore, we discuss numerous variations on this theme.
In this note we prove algebraic independence results for the values of a special class of Mahler functions. In particular, the generating functions of Thue-Morse, regular paperfolding and Cantor sequences belong to this class, and we obtain the algebraic independence of the values of these functions at every non-zero algebraic point in the open unit disk. The proof uses results on Mahler's method.
Transcendence of multivariate Mahler functions and algebraic relations between their values
研究多元马勒函数的函数性质，证明其在特定区域的解析性与亚纯性及有理 - 超越二分法，加强相关提升定理，能计算函数值间代数关系，还证明了正则点情况下的多元下降定理。
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…
The main purpose of this article is to provide new results on algebraic independence of values of Mahler functions and their generalizations. Simultaneously, we establish new measures of algebraic independence for these values. Among the other things, we provide a measure of algebraic independence for values of Mahler's functions at complex transcendental points, a result of type which has never appeared in the literature in the past. As an example of application of our new measures of algebraic independence, we prove that a Mahler number does not belong to the class $U$ in Mahler's classifica…
A Liouville-Type Inequality for Values of Mahler M-Functions
本文基于马赫尔Mq函数在公共非零代数点处的值，建立利维勒型不等式，并证明此类值既不是利维勒数也不是U数，解决了该领域长期存在的问题。
In this paper, we estimate the simultaneous approximation exponents of the values of certain Mahler functions. For this we construct Hermite-Padé approximations of the functions under consideration, then apply the functional equations to get an infinite sequence of approximations and use the numerical approximations obtained from this sequence.
In 1990, Ku. Nishioka proved a fundamental theorem for Mahler's method, which is the analog of the Siegel-Shidlovskii theorem for Mahler functions. In this article, we establish a version of the theorem of Ku. Nishioka which is also valid for Mahler systems over function fields of positive characteristic. We follow the approach introduced by Denis in 1999 in a particular case. It is based on an algebraic independence criterion from Philippon. The main motivation of this work is built on the following remarkable fact discovered by Denis. Over function fields of positive characteristic, analogs …
In this paper, we estimate the linear independence measures for the values of a class Mahler functions of degree one and two. For the purpose, we study the determinants of suitable Hermite-Padé approximation polynomials. Based on the non-vanishing of these determinants, we apply the functional equations to get an infinite sequence of approximations which is used to produce the linear independence measures.
In this paper, we give a new proof of a result due to Bezivin that a D-finite Mahler function is necessarily rational. This also gives a new proof of the rational-transcendental dichotomy of Mahler functions due to Nishioka. Using our method of proof, we also provide a new proof of a Polya-Carlson type result for Mahler functions due to Rande; that is, a Mahler function which is meromorphic in the unit disk is either rational or has the unit circle as a natural boundary.
The Mahler measure of a polynomial $P$ in $n$ variables is defined as the mean of $\log|P|$ over the $n$-dimensional torus. For certain polynomials with integer coefficients in two variables the Mahler measure is known to be related to special values of L-functions of arithmetic objects (e.g. Dirichlet characters and elliptic curves over $\mathbb{Q}$). Inspired by work of Deninger Boyd has investigated this relationship numerically. In this paper we reduce some conjectures of Boyd to Beilinson`s conjectures on special values of L-functions. The methods in use are widely of K-theoretical nature…
Mahler measure of some n-variable polynomial families
利用黎曼ζ函数和Dirichlet L-级数的特殊值给出了一类n元多项式族的Mahler测度，推广了\cite{L}的结果.在这项工作中介绍的技术也激励伯努利数和对称函数之间的某些身份。
We study a class of 2-variable polynomials called exact polynomials which contains $A$-polynomials of knot complements. The Mahler measure of these polynomials can be computed in terms of a volume function defined on the vanishing set of the polynomial. We prove that the local extrema of the volume function are on the 2-dimensional torus and give a closed formula for the Mahler measure in terms of these extremal values. This formula shows that the Mahler measure of an irreducible and exact polynomial divided by $π$ is greater than the amplitude of the volume function. We also prove a $K$-theor…
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 $α$ and $β$ be positive real numbers. Let $F(x) \in K[[x^Γ]]$ be a Hahn series. We prove that if $F(x)$ is both $α$-Mahler and $β$-Mahler then it must be a rational function, $F(x) \in K(x)$, assuming some non-degeneracy conditions on $α$ and $β$.
The Mahler measure of a family of polynomials with arbitrarily many variables
本研究针对任意多变量的无限多项式族，通过积分变换与迭代化简为六次单位根处多重对数求值，推导出其Mahler测度的精确公式，得到Riemann zeta函数与导子3的Dirichlet L函数相关值。
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.
Mahler discrete residues and summability for rational functions
本文为有理函数构造Mahler离散留数，证明其完全刻画Mahler可和性，并推广了Chen和Singer的离散留数理论至Mahler情形。
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 $\mathbb{C}$. In this work, we provide a positive answer for this question by showing the existence of uncountable many of these functions.
A purity theorem for Mahler equations
本文提出了一项关于马勒方程的纯度定理，通过分析广义马勒级数的滤波结构，证明了其在解中的传播性质。
Twisted Mahler discrete residues
本文提出整数λ对应的λ-扭曲马勒离散留数概念，证明其构成扭曲马勒可和性问题的完全障碍，并给出其在微分创造望远镜问题、线性马勒方程微分伽罗瓦理论中的初步应用。
Mahler's method and Carlitz logarithm
本文用马勒方法重新证明了卡尔利茨对数在有理函数域上线性无关则代数无关的定理，扩展了Denis的方法。
In 1906, Maillet proved that given a non-constant rational function $f$, with rational coefficients, if $ξ$ is a Liouville number, then so is $f(ξ)$. Motivated by this fact, in 1984, Mahler raised the question about the existence of transcendental entire functions with this property. In this work, we provide an uncountable subset of Liouville numbers for which there exists a transcendental entire function taking this set into the set of the Liouville numbers.
Mahler measures and computations with regulators
在这项工作中，我们应用的技术，在[拉林：代数积分马勒措施]，以研究几个例子的多变量多项式的马勒措施表示的特殊价值观的黎曼ζ函数或狄利克雷L-系列。这些例子可以从监管机构的评估角度来理解。此外，我们应用相同的技术来计算广义马勒措施，在Gon和Oyanagi的意义[Gon，Oyanagi：广义马勒措施和多个正弦函数]
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…
A height gap theorem for coefficients of Mahler functions
本文研究了马勒函数系数的高度增长行为，证明了存在五种不同的增长模式，并揭示了其中的间隙现象，同时推导了k-马勒函数的正则性和自动性条件。
Random walks through the areal Mahler measure: steps in the complex plane
本文通过随机游走和Zeta Mahler函数的面积类比，证明了两个多项式族的面积Mahler测度可表示为标准Mahler测度与Deninger循环体积的线性组合。
We establish a functional identity for Mahler measures of the two-parametric family $P_{a,c}(x,y)=a(x+1/x)+y+1/y+c$. Our result extends an identity proven in a paper of Lalín, Zudilin and Samart. As a by-product, we obtain evaluations of $m(P_{a,c})$ for some algebraic values of $a$ and $c$ in terms of special values of $L$-functions and logarithms. We also give a sufficient condition for validity of a certain identity between the elliptic integrals of the first and the third kind, which implies several identities for $m(P_{a,c})$.
Following K. Mahler's suggestion for further research on intrinsic approximation on the Cantor ternary set, we obtain a Dirichlet type theorem for the limit sets of rational iterated function systems. We further investigate the behavior of these approximation functions under random translations. We connect the information regarding the distribution of rationals on the limit set encoded in the system to the distribution of rationals in reduced form by proving a Khinchin type theorem. Finally, using a result of S. Ramanujan, we prove a theorem motivating a conjecture regarding the distribution o…
This article investigates the Mahler measure of a family of 2-variate polynomials, denoted by $P_d, d\geq 1$, unbounded in both degree and genus. By using a closed formula for the Mahler measure introduced in "Volume function and Mahler measure of exact polynomials" (by Guilloux and Marché), we are able to compute $m(P_d)$, for arbitrary $d$, as a sum of the values of dilogarithm at special roots of unity. We prove that $m(P_d)$ converges and the limit is proportional to $ζ(3)$, where $ζ$ is the Riemann zeta function.
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 $\mathbb{C}$.
Let $M(α)$ denote the Mahler measure of the algebraic number $α$. In a recent paper, Dubickas and Smyth constructed a metric version of the Mahler measure on the multiplicative group of algebraic numbers. Later, Fili and the author used similar techniques to study a non-Archimedean version. We show how to generalize the above constructions in order to associate, to each point in $(0,\infty]$, a metric version $M_x$ of the Mahler measure, each having a triangle inequality of a different strength. We are able to compute $M_x(α)$ for sufficiently small $x$, identifying, in the process, a function…
The asymptotic Mahler measure of Gaussian periods
构造具有特别小Mahler测度的分圆整数（高斯周期）序列，研究其Mahler测度作为导子函数的渐近行为，发现增长率是维数递增的log Calabi-Yau簇族的多元Mahler测度。
The Mahler measure was introduced by Mahler in the study of number theory. It is known that the Mahler measure appears in different areas of mathematics and physics. On the other hand, we have been investigated a new class of zeta functions for various kinds of walks including quantum walks by a series of our previous work on "Zeta Correspondence". The quantum walk is a quantum counterpart of the random walk. In this paper, we present a new relation between the Mahler measure and our zeta function for quantum walks. Firstly we consider this relation in the case of one-dimensional quantum walks…
The distribution of Mahler's measures of reciprocal polynomials
本文研究了复系数有界偶数次互反多项式的Mahler测度的分布。我们发现，与Mahler测度有关的分布函数在[1，\infty）上是一个互反（或反互反）Laurent多项式，在[0，1）上是恒零的.此外，这个劳伦多项式的系数是有理数乘以π的幂。我们是通过计算分布函数的梅林变换得出这一发现的。这种梅林变换是一个偶（或奇）有理函数，其极点位于小整数处，残差是有理数乘以π的幂。我们还使用这种梅林变换表明，一组倒数多项式的体积与复系数，有界度和马勒的措施小于或等于一个是一个有理数倍的幂π。
In 1994, Becker conjectured that if $F(z)$ is a $k$-regular power series, then there exists a $k$-regular rational function $R(z)$ such that $F(z)/R(z)$ satisfies a Mahler-type functional equation with polynomial coefficients where the initial coefficient satisfies $a_0(z)=1$. In this paper, we prove Becker's conjecture in the best-possible form; we show that the rational function $R(z)$ can be taken to be a polynomial $z^γQ(z)$ for some explicit non-negative integer $γ$ and such that $1/Q(z)$ is $k$-regular.
Mahler Measure Variations, Eisenstein Series and Instanton Expansions
本文指出与“模马勒测度”有关的爱森斯坦级数与“非临界弦”的瞬子数之间的一个有趣的反函数关系。在一个配套文件马勒措施有关的二聚体模型。
We present a new method for algebraic independence results in the context of Mahler's method. In particular, our method uses the asymptotic behaviour of a Mahler function $f(z)$ as $z$ goes radially to a root of unity to deduce algebraic independence results about the values of $f(z)$ at algebraic numbers. We apply our method to the canonical example of a degree two Mahler function; that is, we apply it to $F(z)$, the power series solution to the functional equation $F(z)-(1+z+z^2)F(z^4)+z^4F(z^{16})=0$. Specifically, we prove that the functions $F(z)$, $F(z^4)$, $F'(z)$, and $F'(z^4)$ are alg…
The minimal integral Mahler measure of a number field $K$, $M(\mathcal{O}_K)$, is the minimal Mahler measure of a non-torsion primitive element of $\mathcal{O}_K$. Upper and lower bounds, which depend on the discriminant, are known. We show that for cubics, the lower bounds are sharp with respect to its growth as a function of discriminant. We construct an algorithm to compute $M(\mathcal{O}_K)$ for all cubics with absolute value of the discriminant bounded by $N$.
Recent work of Pritsker defines and studies an areal version of the Mahler measure. We further explore this function with a particular focus on the case where its value is small, as this is most relevant to Lehmer's conjecture. In this situation, we provide improvements to two inequalities established in Pritsker's original paper.
Mahler Measure, Eisenstein Series and Dimers
本说明揭示了一个神秘的联系之间的分拆函数的某些二聚体模型的二维环面和$L$ -功能的光谱曲线。它也将某些二聚体模型族中的分拆函数与爱森斯坦级数联系起来。
This paper is concerned with Mahler's method. We study in detail the structure of linear relations between values of Mahler functions at algebraic points. In particular, given a field ${\bf k}$, a Mahler function $f(z)\in{\bf k}\{z\}$, and an algebraic number $α$, $0<\vert α\vert <1$, that is not a pole for $f$, we show that one can always determined whether the number $f(α)$ is transcendental or not. In the latter case, we obtain that $f(α)$ belong to the number fields ${\bf k}(α)$. We also consider some consequences of such results to a classical number theoretical problem: the study of sequ…
Some results on asymptotic versions of Mahler's problems
该研究属于数论领域，证明了两类关于系数几乎全有界的超越函数的结果，推进了Mahler问题渐近版本的相关研究。
Suppose that $F(x)\in\mathbb{Z}[[x]]$ is a Mahler function and that $1/b$ is in the radius of convergence of $F(x)$. In this paper, we consider the approximation of $F(1/b)$ by algebraic numbers. In particular, we prove that $F(1/b)$ cannot be a Liouville number. If $F(x)$ is also regular, we show that $F(1/b)$ is either rational or transcendental, and in the latter case that $F(1/b)$ is an $S$-number or a $T$-number.
E. Maillet proved that the set of Liouville numbers is preserved under rational functions with rational coefficients. Based on this result, a problem posed by Kurt Mahler is to investigate whether there exist entire transcendental functions with this property or not. For large parametrized classes of Liouville numbers, we construct such functions and moreover we show that it can be constructed such that all their derivatives share this property. We use a completely different approach than in a recent paper, where functions with a different invariant subclass of Liouville numbers were construct…
The parametrized family of metric Mahler measures
本文推广了Mahler测度的度量版本$M_t$，利用其给出Lehmer猜想的等价形式，证明该函数由指数函数分段构造，并提出一个关于有理数图像绘制的猜想。
The Mahler measure of exact polynomials and special $L$-values of $K3$ surfaces
本文用Deligne-Beilinson上同调表达任意变量精确多项式的Mahler测度，并推广至四变量情形，联系$K3$曲面在$s=4$的$L$-值；在Beilinson猜想下，将特定多项式测度用zeta函数和模形式$L$-函数表示。
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…
We investigate the continued fraction expansion of the infinite products $g(x) = x^{-1}\prod_{t=0}^\infty P(x^{-d^t})$ where polynomials $P(x)$ satisfy $P(0)=1$ and $°(P)<d$. We construct relations between partial quotients of $g(x)$ which can be used to get recurrent formulae for them. We provide that formulae for the cases $d=2$ and $d=3$. As an application, we prove that for $P(x) = 1+ux$ where $u$ is an arbitrary rational number except 0 and 1, and for any integer $b$ with $|b|>1$ such that $g(b)\neq0$ the irrationality exponent of $g(b)$ equals two. In the case $d=3$ we provide a partial …
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.
In a recent paper the team of Cogdell, Jorgenson and Smajlović develop infinite series representations for the logarithmic Mahler measure of a complex linear form, with 4 or more variables. We establish the case of 3 variables, by bounding an integral with integrand involving the random walk probability density $a\displaystyle\int_0^\infty tJ_0(at) \displaystyle\prod_{m=0}^2 J_0(r_m t)dt$, where $J_0$ is the order zero Bessel function of the first kind, and $a$ and {$r_m$} are positive real numbers. To facilitate our proof we develop an alternative description of the integral's asymptotic beha…
In 1984, Kurt Mahler posed the following fundamental question: How well can irrationals in the Cantor set be approximated by rationals in the Cantor set? Towards development of such a theory, we prove a Dirichlet-type theorem for this intrinsic diophantine approximation on Cantor-like sets, and discuss related possible theorems/conjectures. The resulting approximation function is analogous to that for R^d, but with d being the Hausdorff dimension of the set, and logarithmic dependence on the denominator instead.
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