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H . Niederreiter

2 . 8 . Lemma. Let Pbe a (/, m, s)-net in base b, let E be an elementary interval in base b with V{E) = b ~", where 0 ^u^m — t, and let T be an affine transformation from E onto Г. Then the points ofP that belong to E are transformed by T into a {t,m — м, synet in base b.

Proof Since E can be written as the disjoint union of /?'"-'-" elementary intervals in base b with volume Z>^"'", it follows that E contains exactly Z?'""" points of P. If these points are transformed by Г, we get a point set P' of Z?'"~" points in Г. To prove that P' is a (/, m — Щ 5')-net in base b, we take an elementary interval E' in base b with ViE") = /?'-'"+" and note that for x g £ we have Г(х) e £" if and only if XG r~^ (£"). Now T~^ (£") is an elementary interval in base b with К(Г-Ч^')) = ^'"^ thus T-^E") contains exactly b' points of P. Consequently, E' contains exactly b^ points of P'. П

2 . 9 . Lemma. If 0 ^ t ^m and h ^ 1 are integers, then every {th.mh,synet in base b is a (t,m,synet in base b^.

Proof A {th,mh,synQt in base b contains exactly b"^^ points, which is the required number of points of a (?,m,5)-net in base b^. Furthermore, an elementary interval in base b^ with volume b^^^'^^Hs also an elementary interval in base b, and so it contains exactly b^^ points of a {th,mh,synQt in base b. П

3 . The Discrepancy of Nets

We generalize and improve the upper bounds of Sobol' [29] and Faure [4] for the discrepancy of (t, m, 5)-nets in base b. We denote by [x\ the greatest integer ^ x.

3 . L Theorem. The discrepancy A {b"^) of a (t, m, synet in base b^ 3 satisfies

where q = min(m — t,s — 1).

Proof Let Ai,{t,m,s) denote the right-hand side of (3.1). We fix t ^ 0 and proceed by double induction on ^ ^ 1 and m ^ л First let ^ = 1 and consider an arbitrary m^ t. If an interval / = [0, u), 0 < w ^ 1, is given, we split it up into the disjoint intervals /;, =