Density property of certain sets and their applications

In this paper we show that certain sets are dense in $\mathbb{R}$. We give some applications. For example, we show an analytical proof that $q^{\frac{1}{n}}$, $q$ is a prime number and $e$; are irrational numbers. As another application we show: If $f$ is an locally integrable function on $\mathbb{R}-\{0\}$ satisfying $\int_x ^{px}f(t)dt$ and $\int_x ^{qx}f(t)dt$ are constant with $\frac{\ln p}{\ln q}$ is an irrational number; implies $f(t)=\frac{c}{t}\,\,\ a.e.$, where $c$ is constant; which is already considered in \cite{b1} for the case when $f$ is continuous.