Density property of certain sets and their applications

TL;DR

This paper demonstrates the density of certain sets in real numbers and applies these results to prove the irrationality of specific numbers and characterize functions based on integral conditions.

Contribution

It provides new proofs of irrationality for numbers like $q^{1/n}$ and $e$, and characterizes functions satisfying particular integral equations using density properties.

Findings

Proof that $q^{1/n}$ and $e$ are irrational.

Characterization of functions with specific integral properties.

Establishment of density results for certain sets in $ eal$.

Abstract

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.