1

I want to find the set of parameters $\alpha\in\mathbb{R}$ such that the sequence

$$ x_n=(\alpha \lambda^nv)\,\mathrm{mod}\,1 $$ is dense in the unit square $\lbrack 0,1\rbrack^2$ where $$ \lambda=\frac{3+\sqrt{5}}{2}\qquad\text{and}\qquad v=\begin{pmatrix}\frac{1+\sqrt{5}}{2}\\1\end{pmatrix}. $$ One may exploit that $v$ is the eigenvector corresponding to the eigenvalue $\lambda$ of the matrix $$ A=\begin{pmatrix}2&1\\1&1\end{pmatrix}, $$ i.e. $x_n=(A^n(\alpha v))\,\mathrm{mod}\,1$

I'm thankful for any kind of advice.

lbf_1994
  • 439
  • 2
  • 13

0 Answers0