This paper by H. J. Landau addresses the necessary density conditions for sampling and interpolation of certain entire functions in Euclidean spaces. The work builds on Beurling's earlier problems of balayage and interpolation, and extends them to the context of $ L^2 $ spaces. The key results establish that for a set $ \Lambda $ to be a set of sampling or interpolation for a space $ \mathcal{B}(S) $, it must satisfy certain density conditions related to the measure of $ S $. The paper introduces the concepts of upper and lower uniform densities for a uniformly discrete set $ \Lambda $, defined as the limits of the ratios of the number of points in $ \Lambda $ to the volume of intervals as the interval length tends to infinity. These densities are crucial in determining the sampling and interpolation conditions. The main theorems establish that for a set $ \Lambda $ to be a set of sampling for $ \mathcal{B}(S) $, its lower uniform density must exceed a certain threshold related to the measure of $ S $, and for it to be a set of interpolation, its upper uniform density must be bounded by a similar threshold. These results are derived using eigenvalue analysis and properties of Fourier transforms. The paper also proves Beurling's conjecture regarding the density conditions for balayage, showing that if balayage is possible for a set $ S $ and $ \Lambda $, then $ \Lambda $ contains a uniformly discrete subset with a lower uniform density exceeding a specific value related to the measure of $ S $. The results are presented in the context of $ L^2 $ spaces and involve the use of Fourier transforms, eigenvalue problems, and properties of entire functions. The paper concludes with the establishment of sharp density-measure theorems for sets of sampling and interpolation, and the confirmation of Beurling's conjecture.This paper by H. J. Landau addresses the necessary density conditions for sampling and interpolation of certain entire functions in Euclidean spaces. The work builds on Beurling's earlier problems of balayage and interpolation, and extends them to the context of $ L^2 $ spaces. The key results establish that for a set $ \Lambda $ to be a set of sampling or interpolation for a space $ \mathcal{B}(S) $, it must satisfy certain density conditions related to the measure of $ S $. The paper introduces the concepts of upper and lower uniform densities for a uniformly discrete set $ \Lambda $, defined as the limits of the ratios of the number of points in $ \Lambda $ to the volume of intervals as the interval length tends to infinity. These densities are crucial in determining the sampling and interpolation conditions. The main theorems establish that for a set $ \Lambda $ to be a set of sampling for $ \mathcal{B}(S) $, its lower uniform density must exceed a certain threshold related to the measure of $ S $, and for it to be a set of interpolation, its upper uniform density must be bounded by a similar threshold. These results are derived using eigenvalue analysis and properties of Fourier transforms. The paper also proves Beurling's conjecture regarding the density conditions for balayage, showing that if balayage is possible for a set $ S $ and $ \Lambda $, then $ \Lambda $ contains a uniformly discrete subset with a lower uniform density exceeding a specific value related to the measure of $ S $. The results are presented in the context of $ L^2 $ spaces and involve the use of Fourier transforms, eigenvalue problems, and properties of entire functions. The paper concludes with the establishment of sharp density-measure theorems for sets of sampling and interpolation, and the confirmation of Beurling's conjecture.