Change search
ReferencesLink to record
Permanent link

Direct link
Exponential squared integrability of the discrepancy function in two dimensions
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2009 (English)In: Mathematika, ISSN 0025-5793, Vol. 55, no 1-2, 1-27 p.Article in journal (Refereed) Published
Abstract [en]

Let AN be an N-point set in the unit square and consider the discrepancy function DN(x):= #(AN ∩ [0, x)) - N|(0, x)|, where x = (x1, X2) ∈ [0, 1]2, [0, x) = Πt=1 2[0, xt), and |[0, x)| denotes the Lebesgue measure of the rectangle. We give various refinements of a well-known result of Schmidt [Irregularities of distribution. VII. Acta Arith. 21 (1972), 45-50] on the L∞ norm of DN. We show that necessarily ||DN||exp(L α) ≳ (log N)1-1/α, 2 ≤ α < ∞. The case of α = ∞ is the Theorem of Schmidt. This estimate is sharp. For the digit-scrambled van der Corput sequence, we have ||DN|| exp(L α) ≳ (log N)1-1/α, 2 ≤ α < ∞., whenever N = 2n for some positive integer n. This estimate depends upon variants of the Chang-Wilson - Wolff inequality [S.-Y. A. Chang, J. M. Wilson and T. H. Wolff, Some weighted norm inequalities concerning the Schrödinger operators. Comment. Math. Helv. 60(2) (1985), 217-246]. We also provide similar estimates for the BMO norm of DN.

Place, publisher, year, edition, pages
2009. Vol. 55, no 1-2, 1-27 p.
National Category
URN: urn:nbn:se:kth:diva-152798ScopusID: 2-s2.0-74049140472OAI: diva2:751484

QC 20141001

Available from: 2014-10-01 Created: 2014-10-01 Last updated: 2014-10-01Bibliographically approved

Open Access in DiVA

No full text


Search in DiVA

By author/editor
Parissis, Ioannis
By organisation
Mathematics (Dept.)
In the same journal

Search outside of DiVA

GoogleGoogle Scholar
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

Total: 25 hits
ReferencesLink to record
Permanent link

Direct link