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Optimal lower bounds for discrepancy in Besov spaces with dominating mixedsmoothness are known from the work of Triebel. Hinrichs proved upper bounds inthe plane. In this work we systematically analyse the problem, starting with asurvey of discrepancy results and the calculation of the best known constant inRoth's Theorem. We give a larger class of point sets satisfying the optimalupper bounds than already known from Hinrichs for the plane and solve theproblem in arbitrary dimension for certain parameters considering a celebratedconstructions by Chen and Skriganov which are known to achieve optimal$L_2$-norm of the discrepancy function. Since those constructions are $b$-adic,we give $b$-adic characterizations of the spaces. Finally results forTriebel-Lizorkin and Sobolev spaces with dominating mixed smoothness and forthe integration error are concluded.

Discrepancy and integration in function spaces with dominating mixed smoothness