The first part introduces a mathematical model capable of predicting the performance of an accelerators vacuum system. A coupled balance equation system describes the distribution of the gas dynamics considering impacts of conductance limitations, beam induced effects, thermal outgassing and sticking probabilities of the chamber materials. A new solving algorithm based on sparse matrix representations, is introduced. The model is implemented in a Python environment. A sensitivity analysis, a cross-check with the software Molflow+ and a comparison to readings of the LHC pressure gauges validate the model. A simulation of the vacuum system for the potential future accelerator FCC with 100 km in circumference is shown. The second part of the thesis studies properties of quasi-Monte Carlo methods. Instead of solving a complex integral analytically, its value is approximated by function evaluation at specific points. A good point set is critical for a good result. Continuous curves provide a good tool to define these point sets. The “bounded remainder sets” (BRS) define a measure for the quality of the uniform distribution. The trajectory of a billiard path with an irrational slope is especially well distributed. Certain criteria to the BRS are defined and the distribution error is analysed. The proofs are based on Diophantine approximations of irrational numbers and on the unfolding technique of the billiard path to a straight line in the plane. The third part analyses the distribution of the energy levels of quantum systems. It was stated that the eigenvalues of the energy spectra for almost all integrable quantum systems are uncorrelated and Poisson distributed. The harmonic oscillator presents one counter example to this assertion. A statement is forumlated to describe when the eigenvalues do not follow the poissonian property. The concept of the proofs is based on the analysis of the pair correlations of sequences.