Star discrepancy estimates for digital \((t,m,2)\)-nets and digital \((t,2)\)-sequences over \(\mathbb Z_2\). (English) Zbl 1102.11036

Dick, J.; Kritzer, P.

Acta Math. Hung. 109, No. 3, 239-254 (2005).

The authors prove that for the star discrepancy of digital \((t,m,2)\)-nets over \(\mathbb Z_2\) the upper estimate \(2^mD^*_{2^m}(P)\leqq 2^t((m-t)/3+19/9)\) holds. This is a generalization of the result on \((0,m,2)\)-nets of G. Larcher and F. Pillichshammer [Acta Arith. 106, 379–408 (2003; Zbl 1054.11039)]. It is shown that the constant in the estimate is best possible. An upper bound for the star discrepancy of digital \((t,2)\)-sequences over \(\mathbb Z_2\) is given. This is a generalization of the result on \((0,2)\)-sequences of F. Pillichshammer [Acta Arith. 108, 167–189 (2003; Zbl 1054.11040)].

Reviewer: Yukio Ohkubo (Kagoshima)

MSC:

11K06	General theory of distribution modulo \(1\)
11K38	Irregularities of distribution, discrepancy

Keywords:

digital \((t,m,s)\)-nets; digital \((t,s)\)-sequences; star discrepancy

Citations:

Zbl 1054.11039; Zbl 1054.11040