The asymptotic behaviour of a Sturm-Liouville differential equation with coefficient ${\lambda }^{2}q\left(s\right),s\in [s_0,\infty )$ is investigated, where $\lambda \in ℝ$ and $q\left(s\right)$ is a nondecreasing step function tending to $\infty$ as $s\to \infty$. Let $S$ denote the set of those $\lambda$’s for which the corresponding differential equation has a solution not tending to 0. It is proved that $S$ is an additive group. Four examples are given with $S=\left\{0\right\}$, $S=\mathbb{Z}$, $S=𝔻$ (i.e. the set of dyadic numbers), and $\mathbb{Q}\subset S⫋ℝ$.

We prove: If $$\frac{1}{2}+\sum _{k=1}^n{a}_k\left(n\right)cos\left(kx\right)\ge 0\text{for}\text{all}x\in [0,2\pi ),$$ then $$1-a_k\left(n\right)\ge \frac{1}{2}\frac{k^2}{n^2}\text{for}k=1,\cdots ,n.$$ The constant $1/2$ is the best possible.