On the cube-discrepancy of Kronecker-sequences (Q2277511)

For an infinite sequence \(x_ 1,x_ 2,..\). in \([0,1)^ s\) let \(W_ N^{(r)}\) denote the discrepancy of the first N members of the sequence with respect to cubes in arbitrary position in \([0,1)^ s\) and with side-length at most r. For an s-tupel of reals \((\alpha_ 1,...,\alpha_ s)\) let \(x_ n:=(\{n\alpha_ 1\},...,\{n\alpha_ s\})\); \(n=1,2,..\). be the s-dimensional Kronecker-sequence. Partly best possible estimates of \(W_ N^{(r)}\) for the Kronecker-sequence in terms of approximability by rationals of the s-tupel \((\alpha_ 1,...,\alpha_ s)\) are given.