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The $L_p$-discrepancy is a quantitative measure for the irregularity ofdistribution modulo one of infinite sequences. In 1986 Proinov proved for all$p>1$ a lower bound for the $L_p$-discrepancy of general infinite sequences inthe $d$-dimensional unit cube, but it remained an open question whether thislower bound is best possible in the order of magnitude until recently. In 2014Dick and Pillichshammer gave a first construction of an infinite sequence whoseorder of $L_2$-discrepancy matches the lower bound of Proinov. Here we give acomplete solution to this problem for all finite $p > 1$. We consider so-calledorder $2$ digital $(t,d)$-sequences over the finite field with two elements andshow that such sequences achieve the optimal order of $L_p$-discrepancysimultaneously for all $p \in (1,\infty)$.

Optimal $L_p$-discrepancy bounds for second order digital sequences