JMO 1996 Problem 5

Let $q$ be a real number with $\frac{1+\sqrt{5}}{2}<q<2$. If a positive integer $n$ is represented in binary system as $2^k+a_{k-1}2^{k-1}+\cdots +2a_1+a_0$, where $a_i\in\{0,1\}$, define $$p_n=q^k+a_{k-1}q^{k-1}+\cdots +qa_1+a_0.$$

Prove that there exist infinitely many positive integers $k$ with the property that there is no $l\in\mathbb{N}$ such that $p_{2k}<p_l< p_{2k+1}$.