The extreme and L2 discrepancies of some plane sets

L 2 Discrepancy of Two-Dimensional Digitally Shifted Hammersley Point Sets in Base b

“…Once again, we need only two permutations but our results are valid for arbitrary bases whereas Halton & Zaremba and Kritzer & Pillichshammer deal only with base 2. We also remark that in base 2, shift and swap is the same permutation, so that [8] fully generalizes the results of [9] (for L 2 discrepancy) and [10] from base 2 to base b. Now, after White who needs b permutations and Faure & Pillichshammer who need two, the question arises if only one permutation is enough to get the same property, i.e., the best order of L 2 discrepancy.…”

“…Exact formulas for the L 2 discrepancy of the classical two-dimensional Hammersley point set H id b,n in base b have been proved by Vilenkin [17], Halton & Zaremba [9] and Pillichshammer [13] in base b = 2 and by White [18] and Faure & Pillichshammer [8] for arbitrary bases. These results show that the classical Hammersley point set cannot achieve the best possible order of L 2 discrepancy with respect to Roth's general lower bound (1).…”

“…First results were available in base b = 2: Let id be the identity and id 1 (k) := k + 1 (mod 2) be the digital shift in base 2; then Halton & Zaremba [9] and later, in a much more general form, Kritzer & Pillichshammer [10] gave sequences of permutations Σ ∈ {id, id 1 } n (although they did not use this terminology), for which the generalized Hammersley point set H Σ 2,n in base 2 achieves the best possible order of L 2 discrepancy in the sense of Roth (1). For more detailed results we refer to [10].…”

“…Results for arbitrary bases were first given by White [18] who generalized the result from [9] in a certain way. He considered sequences Σ of the form…”

“…Setting b = 2 in this formula gives the same sequence as in [9] and the simplest sequence in [10], that is Σ = (id 0 , id 1 , id 0 , id 1 , . .…”