This thesis contains the proof of the so-called Bourgain’s simplex theorem. The theorem states that, for a given \(n\)-simpleks \(\Delta\) in \(\mathbb{R}^{n+1}\) and for a given measurable set of positive upper Banach density \(A\) \( \subseteq \mathbb{R}^{n+1}\), there exists a real number \( \lambda_0\), which depends on \(\Delta\) and A, such that for every real number \(\lambda \geq \lambda_0\), the set A contains the vertices of an isometric copy of \(\lambda \Delta\). The proof used techniques from elementary Fourier analysis. Therefore the first chapter is dedicated to a recollection of basic tools from analysis encountered during the undergraduate studies and to basic results concerning the Fourier transform and Gaussian functions. The second chapter contains the precise formulation of the theorem, as well as introduction to spherical measures, and counting forms. Finally, in the third chapter, complete proof of Bourgain’s theorem is given, by decomposing the counting form into a structured part, an error part, and a uniform part.