75 research outputs found
On the Small Ball Inequality in all dimensions
No full textAbstractLet hR denote an L∞ normalized Haar function adapted to a dyadic rectangle R⊂[0,1]d. We show that for choices of coefficients α(R), we have the following lower bound on the L∞ norms of the sums of such functions, where the sum is over rectangles of a fixed volume:nd−12−η‖∑|R|=2−nα(R)hR(x)‖L∞([0,1]d)≳2−n∑|R|=2−n|α(R)|,for some0<η<12. The point of interest is the dependence upon the logarithm of the volume of the rectangles. With n(d−1)/2 on the left above, the inequality is trivial, while it is conjectured that the inequality holds with n(d−2)/2. This is known in the case of d=2 [Michel Talagrand, The small ball problem for the Brownian sheet, Ann. Probab. 22 (3) (1994) 1331–1354, MR 95k:60049], and a recent paper of two of the authors [Dmitriy Bilyk, Michael T. Lacey, On the Small Ball Inequality in three dimensions, Duke Math. J., (2006), in press, arXiv: math.CA/0609815] proves a partial result towards the conjecture in three dimensions. In this paper, we show that the argument of [Dmitriy Bilyk, Michael T. Lacey, On the Small Ball Inequality in three dimensions, Duke Math. J., (2006), in press, arXiv: math.CA/0609815] can be extended to arbitrary dimension. We also prove related results in the subjects of the irregularity of distribution, and approximation theory. The authors are unaware of any prior results on these questions in any dimension d⩾4
Algorithms and Complexity for Continuous Problems (Dagstuhl Seminar 23351)
No full textThe Dagstuhl Seminar 23351 was held at the Leibniz Center for Informatics, Schloss Dagstuhl, from August 27 to September 1, 2023. This event was the 14th in a series of Dagstuhl Seminars, starting in 1991. During the seminar, researchers presented overview talks, recent research results, work in progress and open problems. The first section of this report describes the goal of the seminar, the main seminar topics, and the general structure of the seminar. The third section contains the abstracts of the talks given during the seminar and the forth section the problems presented at the problem session
Algorithms and Complexity for Continuous Problems (Dagstuhl Seminar 19341)
No full textFrom 18.08. to 23.08.2019, the Dagstuhl Seminar 19341 Algorithms and Complexity for Continuous Problems was held in the International Conference and Research Center (LZI), Schloss Dagstuhl. During the seminar, participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar can be found in this report. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available
Discrepancy theory Radon series on computational and applied mathematics ;, 26./ edited by Dmitriy Bilyk, Josef Dick, Friedrich Pillichshammer
No full textIncludes bibliographical referencesThe contributions in this book focus on a variety of topics related to discrepancy theory, comprising Fourier techniques to analyze discrepancy, low discrepancy point sets for quasi-Monte Carlo integration, probabilistic discrepancy bounds, dispersion of point sets, pair correlation of sequences, integer points in convex bodies, discrepancy with respect to geometric shapes other than rectangular boxes, and also open problems in discrepany theory1 online resource
Fourier analytic techniques for lattice point discrepancy
No full textCounting integer points in large convex bodies with smooth boundaries containing isolated flat points is oftentimes an intermediate case between balls (or convex bodies with smooth boundaries having everywhere positive curvature) and cubes (or convex polytopes). In this paper, we provide a detailed description of several discrepancy problems in the particular planar case where the boundary coincides locally with the graph of the function R ∋ t -> |t|^γ, with γ > 2. We consider both integer points problems and irregularities of distribution problems. The above “restriction” to a particular family of convex bodies is compensated by the fact that many proofs are elementary.
The paper is entirely self-contained
AMS Sectional Meeting
No full textGeorgia Southern faculty member Alexander Stokolos co-edited Recent Advances in Harmonic Analysis and Applications: In Honor of Konstantin Oskolkov in collaboration with non-faculty members Laura De Carli, Dmitriy Bilyk, Alexander Petukhov, and Brett D. Wick.
Book Summary: Recent Advances in Harmonic Analysis and Applications is dedicated to the 65th birthday of Konstantin Oskolkov and features contributions from analysts around the world.
The volume contains expository articles by leading experts in their fields, as well as selected high quality research papers that explore new results and trends in classical and computational harmonic analysis, approximation theory, combinatorics, convex analysis, differential equations, functional analysis, Fourier analysis, graph theory, orthogonal polynomials, special functions, and trigonometric series.
Numerous articles in the volume emphasize remarkable connections between harmonic analysis and other seemingly unrelated areas of mathematics, such as the interaction between abstract problems in additive number theory, Fourier analysis, and experimentally discovered optical phenomena in physics. Survey and research articles provide an up-to-date account of various vital directions of modern analysis and will in particular be of interest to young researchers who are just starting their career. This book will also be useful to experts in analysis, discrete mathematics, physics, signal processing, and other areas of science
Cyclic Shifts of the Van Der Corput Set
No full textIn 1980, K. Roth showed that the expected value of the L2 discrepancy of the cyclic shifts of the N-point van der Corput set is bounded by a constant multiple of √logN, thus guaranteeing the existence of a shift with asymptotically minimal L2 discrepancy. In the present paper, we construct a specific example of such a shift
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