Description
A certain type of approximation by polynomials with algebraic coefficients
Let $N\in\mathbb{N}$ be an integer and $\mathcal{A}=\left\{q_1,....,q_N\right\}$ be a set of algebraic numbers. Given $k\in\mathbb{N}$ call $\mathcal{P}_{\mathcal{A},k}$ the set of polynomials of degree smaller than $k$ and coefficient in $\mathcal{A}$ by $\mathcal{P}_{\mathcal{A},k}$ and $\mathcal{P}_{\mathcal{A}}$ the collection of every polynomials with coefficient in $\mathcal{A}$, that is \begin{align}
\mathcal{P}{\mathcal{A},k}=\left{P(X)=\sum{i=0}^k a_i X^i, a_i \in\mathcal{A}\right}\text{ and }\mathcal{P}{\mathcal{A}}=\bigcup{k\geq 0}\mathcal{P}_{\mathcal{A},k}.
\end{align}
Given $t\in(0,1)$ a natural question is to investigate the Hausdorff dimension of real numbers approximable at a given rate by elements $P(t)$ where $P\in \mathcal{P}_{\mathcal{A}},$ i.e., determining for every mapping $\phi:\mathbb{N}\to \mathbb{R}_+,$ $$\qquad \qquad \dim_H W_{\mathcal{A},t}(\phi)=\left\{x\in\mathbb{R} : \ \vert x-P(t)\vert\leq \phi(\deg(P))\text{ for infinitely many }P\in\mathcal{P}_{\mathcal{A}}\right\}.
$$
For instance, a consequence of the mass transference principle of Beresnevich and Velani shows that for $\mathcal{A}_1=\left{0,\frac{2}{3}\right}$, $t_1=\frac{1}{3}$, $\delta\geq 1,$ $\phi(n)=t_1^{n\delta},$ one has
$$\dim_H
W_{\mathcal{A}_1,t_1}(\phi)=\frac{\log 2}{\delta \log 3}.$$
In comparison, some quick calculation shows that for $\mathcal{A}_2=\left{\frac{1}{7},\frac{3}{7},\frac{5}{7}\right}$, $t_2=\frac{1}{7}$, $\delta> 1,$ $\phi(n)=t_2^{n\delta},$ one has
$$\dim_H
W_{\mathcal{A}_2,t_2}(\phi)=0.$$
In this talk, we provide some general results regarding this problem and some partial results aiming at describing the possible behaviors one can encounter. We will more particularly show that the dichotomy between the two cases mentioned occurs when $t$ is the inverse of a Pisot number and the coefficient are in $\mathbb{Z}[t].$ In particular, the results presented will feature new techniques regarding the mass transference principle when the reference measure is not Lebesgue and the sequence of balls we consider overlaps substantially.
