Can every algebraic integer of degree $3$ be approximated by a quotient of linearly recurrent integer sequences of degree $3$?

Solution 1:

Assume that the polynomial $p(x)$ has three real roots $\alpha_1<\alpha_2<\alpha_3$. Let $r\in\Bbb Q$ such that $\frac{\alpha_1+\alpha_2}2<r<\alpha_2$. Then the largest root, in absolute value, of $q(x)=x^3p(r+1/x)$ is $\beta$ satisfying $\alpha_2=r+1/\beta$.

Write $r=u/v$ with $u,v\in\Bbb Z$ and $v>0$. Then $v^3q(x)$ is an integer coefficients polynomial with leading term $v^3p(0)x^3$. Consequently, $vp(0)\beta$ is the largest root, in absolute value, of a integer coefficients monic polynomial.

A stated in OP, there exists linear recurrence sequences $p_n,q_n$ of degree $3$ (satisfying the same linear recurrence) such that $p_n/q_n\to vp(0)\beta$. Consequently, $$\frac{p'_n}{q'_n}\xrightarrow{n\to\infty} r+1/\beta=\alpha_2$$ where \begin{align} p'_n&=up_n+v^2p(0)q_n\\ q'_n&=vp_n \end{align} which are linear recurrence sequences of degree 3.

As example, $1+2\cos(2\pi/7)$ is the largest root, in absolute value, of the polynomial $x^3-2x^2-x+1$, hence $p_n/q_n\to\cos(2\pi/7)$ where \begin{align} p_{n+3}&=2p_{n+2}+p_{n+1}-p_n&p_0&=b-a&p_1&=c-b&p_2&=c+b-a\\ q_{n+3}&=2q_{n+2}+q_{n+1}-q_n&q_0&=2a&q_1&=2b&p_3&=2c \end{align} for almost every choice of $a,b,c$.