Mahler measure under variations of the base group

An asymptotic approach in Mahler's method

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.

A sequential view of self--similar measures, or, What the ghosts of Mahler and Cantor can teach us about dimension

We show that missing $q$-ary digit sets $F\subseteq[0,1]$ have corresponding naturally associated countable binary $q$-automatic sequence $f$. Using this correspondence, we show that the Hausdorff dimension of $F$ is equal to the base-$q$ logarithm of the Mahler eigenvalue of $f$. In addition, we demonstrate that the standard mass distribution $ν_F$ supported on $F$ is equal to the ghost measure $μ_f$ of $f$.

Mahler's method in several variables I: The theory of regular singular systems

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.

通过缺失数据恢复方法寻找Mahler测度的新极限点

Finding New Limit Points of Mahler Measure by Methods of Missing Data Restoration

本文采用缺失数据恢复方法，扩展了单变量多项式Mahler测度已知极限点的列表，进一步推进了该领域的研究。

Exercising in complex Mahler measures: diamonds are not forever

Recently, Hang Liu and Hourong Qin came up with a numerical observation about the relation between the Mahler measures of one hyperelliptic and two elliptic families. The discoverers foresee a proof of the identities "by extending ideas in" two papers of Matilde Lal\'ın and Gang Wu, the ideas based on a theorem of Spencer Bloch and explicit diamond-operation calculations on the underlying curves. We prove the relation using the already available diamond-free methodology. While finding such relations for the Mahler measures remains an art, proving them afterwards is mere complex (analysis) exer…

On a problem posed by Mahler

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…

Short walk adventures

We review recent development of short uniform random walks, with a focus on its connection to (zeta) Mahler measures and modular parametrisation of the density functions. Furthermore, we extend available "probabilistic" techniques to cover a variation of random walks and reduce some three-variable Mahler measures, which are conjectured to evaluate in terms of $L$-values of modular forms, to hypergeometric form.

将马赫勒测度与狄利克雷L值联系起来：陈伯格猜想的新证据

Relating Mahler measures and Dirichlet $L$-values: new evidence for Chinburg's conjectures

本文通过新的数值例子验证了陈伯格猜想，展示了8个强形式和18个弱形式的实例，基于Boyd和Rodriguez-Villegas的工作，证明了允许环面系数的弱猜想。

有限图的 Z-覆盖中的生成树与马勒测度

Spanning trees in $\mathbb{Z}$-covers of a finite graph and Mahler measures

本文利用 Artin-Ihara L-函数在 u=1 处的特殊值定义 Ihara 多项式，并用其马勒测度及 p-进马勒测度刻画 Z-覆盖的有限中间图生成树数目的渐近增长和 p-进赋值，统一并推广了循环图、I-图及 Fibonacci 数赋值等已有结果。

机器学习博伊德（Boyd）马勒测度猜想的算术性质

Machine learning the arithmetic of Boyd's Mahler measure conjectures

该研究利用机器学习与统计分析，探究博伊德马勒测度猜想中有理数$r_k$的算术性质，重现关键观测并发现素数2、3下的特殊算术结构。

The range of multiplicative functions on C[x], R[x] and Z[x]

Mahler's measure is generalized to create the class of {\it multiplicative distance functions}. These functions measure the complexity of polynomials based on the location of their zeros in the complex plane. Following work of S.-J. Chern and J. Vaaler in \cite{chern-vaaler}, we associate to each multiplicative distance function two families of analytic functions which encode information about its range on \C[x] and \R[x]. These {\it moment functions} are Mellin transforms of distribution functions associated to the multiplicative distance function and demonstrate a great deal of arithmetic st…

Random Walks, Spectral Gaps, and Khintchine's Theorem on Fractals

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…