Henri Faure: On the star and the $L_2$ discrepancies of digital (0,1)-sequences in prime base

We give explicit formulas for the star and $L_2$ discrepancies of digital $(0,1)$-sequences in arbitrary prime bases, generated by arbitrary non singular upper triangular matrices. The proofs are based on a compact explicit formula for the discrepancy function involving generalizations of the distance to the nearest integer we have already considered in preceding studies. In the special case of non singular upper triangular matrices, this study generalizes to bases $b$ previous results of Larcher and Pillichshammer in base 2. As another consequence of our compact formula, we get also simple explicit formulas for the extreme discrepancy and the diaphony of the same sequences.

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