Dmitriy Bilyk

Exponential squared integrability of the discrepancy function in two dimensions

Let A_N be an N-point distribution in the unit square in the Euclidean plane. We consider the Discrepancy function D_N(x) in two dimensions with respect to rectangles with lower left corner anchored at the origin and upper right corner at the point x. This is the difference between the actual number of points of A_N in such a rectangle and the expected number of points - N x_1x_2 - in the rectangle. We prove sharp estimates for the BMO norm and the exponential squared Orlicz norm of D_N(x). For example we show that necessarily ||D_N||_(expL^2) >c(logN)^(1/2) for some aboslute constant c>0. On the other hand we use a digit scrambled version of the van der Corput set to show that this bound is tight in the case N=2^n, for some positive integer n. These results unify the corresponding classical results of Roth and Schmidt in a sharp fashion.

Fibonacci sets and symmetrization in discrepancy theory

We study the Fibonacci sets from the point of view of their quality with respect to discrepancy and numerical integration. Letfbng 1=0 be the sequence of Fibonacci numbers. The bn-point Fibonacci setFn [0; 1] 2 is dened as Fn :=f(=b n;fb n 1=bng)g bn =1 ; wherefxg is the fractional part of a number x2 R. It is known that cubature formulas based on Fibonacci setFn give optimal rate of error of numerical integration for certain classes of functions with mixed smoothness. We give a Fourier analytic proof of the fact that the symmetrized Fibonacci set F 0 n = Fn[ f(p1; 1 p2) : (p1;p2) 2 Fng has asymptotically minimal L2 discrepancy. This approach also yields an exact formula for this quantity, which allows us to evaluate the constant in the discrepancy estimates. Numerical computations indicate that these sets have the smallest currently known L2 discrepancy among two-dimensional point sets. We also introduce quartered Lp discrepancy which is a modication of the Lp discrepancy symmetrized with respect to the center of the unit square. We prove that the Fibonacci setFn has minimal in the sense of order quartered Lp discrepancy for all p 2 (1;1). This in turn implies that certain twofold symmetrizations of the Fibonacci setFn are optimal with respect to the standard Lp discrepancy.

One-bit sensing, discrepancy and Stolarsky's principle

A sign-linear one bit map from the

Roth’s Orthogonal Function Method in Discrepancy Theory and Some New Connections

d

The Supremum Norm of the Discrepancy Function: Recent Results and Connections

-dimensional sphere

Distributional estimates for the bilinear Hilbert transform

\mathbb S ^{d}

The L2 Discrepancy of Irrational Lattices

to the

CYCLIC SHIFTS OF THE VAN DER CORPUT SET

n

BMO and exponential Orlicz space estimates of the discrepancy function in arbitrary dimension

-dimensional Hamming cube

Однобитовые измерения, дискрепанс и принцип Столярского@@@One bit sensing, discrepancy, and Stolarsky principle

H^n= \{ -1, +1\} ^{n}