The present thesis addresses one of the most fundamental problems related to Monte-Carlo and Quasi-Monte Carlo methods: the near-optimal way to diminish the variance in Monte-Carlo Integration. We propose to study in depth several innovative approaches. A new theoretical framework will be established; a large spectrum of extremely tangible applications, and the heart of many computer graphics or numerical simulation applications, will greatly benefit from the results of this research. The proposed approach is based on self- similar systems generated by substitution rules. Radical-invertible number systems in non-integer bases will be associated with the system.
Based on this core theory, several approaches will be explored in depth. The first approach is purely geometrical, based on the theory of aperiodic tiling. This approach will further develop and improve a novel approach, based on Penrose aperiodic tilings, proposed in 2004. In the present project, we expect to extend our prior approach to multi-dimensional case, and to better control spectral properties of the tilings. The second approach is combinatorial; it follows a recent work performed by Ostromoukhov, in which a considerable improvement of asymptotic terms of extreme and star discrepancies in dimension one has been achieved. These extreme and star discrepancies are the best asymptotic discrepancies known today. In the current proposal, we expect to extend the methodology of combinatorial search for possible permutation, to more general cases of non-integer-based number systems. The third approach is algebraic; it is based on digital scrambling of our generalized radical invertible non-integer-based number systems. We expect to improve the variance of similar known approaches based on generalized Halton, generalized Hammersley, or generalized Niederreiter multidimensional sequences