11 problems

Let $k$ and $l$ be multiplicatively independent positive integers. Let $F\in K[[x]]$ be a power series over a number field $K$ that is both $k$-Mahler and $l$-Mahler. Loxton and Va…

Becker's conjecture. If $F(z)$ is a $k$-regular power series, then there exists a nonzero $k$-regular rational function $R(z)$ such that $F(z)/R(z)$ satisfies a Mahler-type functio…

Let $\mathbf M$ be the set of values under consideration in the paper. For a complex number $\xi$ and a positive integer $d$, define $w_d(\xi)$ as the supremum of the real numbers…

Let $T_p(z)$ denote the valuation generating function associated with a prime $p$. Let $p_1,p_2,\dots,p_m$ be distinct primes, and let $\alpha_1,\alpha_2,\dots,\alpha_m$ be non-zer…

Let $\alpha$ and $\beta$ be irrational and multiplicatively independent. Let $F(x)$ be a Hahn series over $\mathbb{R}$ that is both $\alpha$-Mahler and $\beta$-Mahler. Irrational M…

Badziahin's conjecture. Then $\mu(g_{\mathbf a}(b))=2$ for every $\mathbf a\in\mathbb Z^2$, except in the following three families:

Let $g_{\mathbf{u}}(x)$ be the Mahler function considered in the paper, with parameter $\mathbf{u}=(u,v)\in\mathbb{Z}^2$. A real number is badly approximable if its continued-fract…

Let $k\geq 2$, let $\mathpzc{k}$ be the coefficient field, and let $f(z)\in\mathpzc{k}[[z]]$ be a Mahler power series. A sequence of rational functions is calm in the sense defined…

Let $k\geq 2$, let $\mathpzc{k}$ be the coefficient field, and let $f(z)\in\mathpzc{k}[[z]]$ satisfy the Mahler equation … A sequence of coefficients is precalm in the sense define…

Let $k\geq 2$, let $\mathpzc{k}$ be the coefficient field, and let $f(z)\in\mathpzc{k}[[z]]$. A power series is $k$-Dumas if it satisfies a Mahler equation whose rational-function…

Let $f(x)$ be a polynomial over a finite field, let $p$ be the relevant prime, and let $g_{f(x),p}(z)$ denote the generating function associated with the coefficient sequence for p…