This paper by H. J. Landau addresses the necessary density conditions for sampling and interpolation of certain entire functions in Euclidean spaces. The author builds on the work of Arne Beurling, who posed and discussed two problems: balayage and interpolation. Balayage involves the possibility of representing a function on a set $S$ by a function supported on a subset $\Lambda$ of a locally compact abelian group $G$. Interpolation is the dual problem, where the goal is to recover a function on $S$ from its values on $\Lambda$. Landau focuses on the case where $G$ is the real line and $S$ is a single interval. He provides explicit solutions to these problems and introduces the concepts of uniformly discrete sets and sets of sampling and interpolation. The main results establish density conditions for these sets, showing that for a set $S$ and a set of sampling or interpolation $\Lambda$, the number of points in $\Lambda$ within a ball of radius $r$ must satisfy certain inequalities involving the measure of $S$ and the density of $\Lambda$. The paper also explores the general case in higher dimensions, proving asymptotic versions of the density conditions. Key theorems include: 1. **Theorem 1**: For a set $S$ consisting of a finite number of intervals in $\mathbb{R}^1$ and a set of sampling $\Lambda$, the lower density of $\Lambda$ must satisfy a specific inequality. 2. **Theorem 2**: For a set $S$ consisting of a finite number of intervals in $\mathbb{R}^1$ and a set of interpolation $\Lambda$, the upper density of $\Lambda$ must satisfy a specific inequality. 3. **Theorem 3**: For a set $S$ and a set of sampling $\Lambda$, the lower density of $\Lambda$ must be at least a certain value. 4. **Theorem 4**: For a set $S$ and a set of interpolation $\Lambda$, the upper density of $\Lambda$ must be at most a certain value. These results are used to prove Beurling's conjecture regarding the density conditions for balayage, showing that if balayage is possible for a set $S$ and $\Lambda$, then $\Lambda$ must contain a uniformly discrete subset with a specific density condition. The paper concludes with a detailed proof of this conjecture.This paper by H. J. Landau addresses the necessary density conditions for sampling and interpolation of certain entire functions in Euclidean spaces. The author builds on the work of Arne Beurling, who posed and discussed two problems: balayage and interpolation. Balayage involves the possibility of representing a function on a set $S$ by a function supported on a subset $\Lambda$ of a locally compact abelian group $G$. Interpolation is the dual problem, where the goal is to recover a function on $S$ from its values on $\Lambda$. Landau focuses on the case where $G$ is the real line and $S$ is a single interval. He provides explicit solutions to these problems and introduces the concepts of uniformly discrete sets and sets of sampling and interpolation. The main results establish density conditions for these sets, showing that for a set $S$ and a set of sampling or interpolation $\Lambda$, the number of points in $\Lambda$ within a ball of radius $r$ must satisfy certain inequalities involving the measure of $S$ and the density of $\Lambda$. The paper also explores the general case in higher dimensions, proving asymptotic versions of the density conditions. Key theorems include: 1. **Theorem 1**: For a set $S$ consisting of a finite number of intervals in $\mathbb{R}^1$ and a set of sampling $\Lambda$, the lower density of $\Lambda$ must satisfy a specific inequality. 2. **Theorem 2**: For a set $S$ consisting of a finite number of intervals in $\mathbb{R}^1$ and a set of interpolation $\Lambda$, the upper density of $\Lambda$ must satisfy a specific inequality. 3. **Theorem 3**: For a set $S$ and a set of sampling $\Lambda$, the lower density of $\Lambda$ must be at least a certain value. 4. **Theorem 4**: For a set $S$ and a set of interpolation $\Lambda$, the upper density of $\Lambda$ must be at most a certain value. These results are used to prove Beurling's conjecture regarding the density conditions for balayage, showing that if balayage is possible for a set $S$ and $\Lambda$, then $\Lambda$ must contain a uniformly discrete subset with a specific density condition. The paper concludes with a detailed proof of this conjecture.