The authors study point distributions which possess the structure of finite abelian groups with respect to certain \(p\)-ary arithmetic operations. Let \(\mathcal{D}_N\) be a distribution of \(N>1\) points in the \(d\)-dimensional unit cube \(U^d\). The local discrepancy \(\mathcal{L}[\mathcal{D}_N;Y]\), \(Y=(y_1,\ldots,y_d)\in U^d,\) is defined by \[ \mathcal{L}[\mathcal{D}_N;Y]=\text{card}\{\mathcal{D}_N\cap B_Y\}-N \text{vol}\;B_Y, \] where \(B_Y=[0,y_1)\times\ldots \times[0,y_d)\subset U^d\) is a rectangular box of volume vol \(B_Y=y_1\ldots y_d\). For \(1\leq q<\infty\) the \(L^q\)-discrepancy is defined by \[ \mathcal{L}_q[\mathcal{D}_N]=\left(\int_{U^q}| \mathcal{L}[\mathcal{D}_N;Y]| ^q\,dY\right)^{1/q}. \] By classical results of K. F. Roth [Mathematika 1, 73–79 (1954; Zbl 0057.28604)] (for \(q\geq 2)\) and by W. M. Schmidt [Number Theory and Algebra; Collect. Pap. dedic. H. B. Mann, A. E. Ross, O. Taussky-Todd, 311–329 (1977; Zbl 0373.10020)] for \(d>1\) point distributions always have intrinsic irregularities which can be expressed by the following general lower bound \[ \mathcal{L}_q[\mathcal{D}_N]\gg(\log N)^{\frac{d-1}{2}}; \] for more details concerning irregularities of distributions see the monographs by J. Beck and W. W. L. Chen [Irregularities of distribution. Cambridge Tracts in Mathematics, 89. (Cambridge) etc.: Cambridge University Press (1987; Zbl 0617.10039)], by M. Drmota and R. F. Tichy [Sequences, discrepancies and applications. Lecture Notes in Mathematics. 1651. Berlin: Springer (1997; Zbl 0877.11043)], or by J. Matousek [Geometric discrepancy. An illustrated guide. Algorithms and Combinatorics. 18. Berlin: Springer (1999; Zbl 0930.11060)]. The lower bound (1) is best possible (apart from the value of the implied constant). For suitable point distributions in dimensions \(d=2\) and \(d=3\) the converse inequality was shown by Davenport (1956), Roth (1979, 1980) and by Frolov (1980). For \(d\geq3\) the “constructions” of these low discrepancy point sets depend on probabilistic arguments.
Recently, W. W. L. Chen and M. M. Skriganov [J. Reine Angew. Math. 545, 67–95 (2002; Zbl 1083.11049)] developed an explicit deterministic construction of point sets in arbitrary dimensions with optimal \(L^2\)-discrepancy. In the present paper the explicit construction problem is solved completely (for even \(q=2t\)) by presenting a suitable point distribution in \(U^d\) satisfying \[ \mathcal{L}_{2t}[\mathcal{D}_N]\ll(\log N)^{\frac{d-1}{2}}. \] The proof depends on \(p\)-adic harmonic analysis and Walsh series expansions.