results for au:Bilyk_D in:math

- It is well-known that for every $N \geq 1$ and $d \geq 1$ there exist point sets $x_1, \dots, x_N \in [0,1]^d$ whose discrepancy with respect to the Lebesgue measure is of order at most $(\log N)^{d-1} N^{-1}$. In a more general setting, the first author proved together with Josef Dick that for any normalized measure $\mu$ on $[0,1]^d$ there exist points $x_1, \dots, x_N$ whose discrepancy with respect to $\mu$ is of order at most $(\log N)^{(3d+1)/2} N^{-1}$. The proof used methods from combinatorial mathematics, and in particular a result of Banaszczyk on balancings of vectors. In the present note we use a version of the so-called transference principle together with recent results on the discrepancy of red-blue colorings to show that for any $\mu$ there even exist points having discrepancy of order at most $(\log N)^{d-\frac12} N^{-1}$, which is almost as good as the discrepancy bound in the case of the Lebesgue measure.
- Dec 28 2016 math.CA arXiv:1612.08442v1We study energy integrals and discrete energies on the sphere, in particular, analogs of the Riesz energy with the geodesic distance in place of Euclidean, and observe that the range of exponents for which the uniform distribution optimizes such energies is different from the classical case. We also obtain a general form of the Stolarsky principle, which relates discrete energies to certain $L^2$ discrepancies. This leads to new proofs of discrepancy estimates, as well as the sharp asymptotics of the difference between optimal discrete and continuous energies in the geodesic case.
- Nov 15 2016 math.CA arXiv:1611.04420v1The classical Stolarsky invariance principle connects the spherical cap $L^2$ discrepancy of a finite point set on the sphere to the pairwise sum of Euclidean distances between the points. In this paper we further explore and extend this phenomenon. In addition to a new elementary proof of this fact, we establish several new analogs, which relate various notions of discrepancy to different discrete energies. In particular, we find that the hemisphere discrepancy is related to the sum of geodesic distances. We also extend these results to arbitrary measures on the sphere and arbitrary notions of discrepancy and apply them to problems of energy optimization and combinatorial geometry and find that, surprisingly, the geodesic distance energy behaves differently than its Euclidean counterpart.
- We obtain mproved bounds for one bit sensing. For instance, let $ K_s$ denote the set of $ s$-sparse unit vectors in the sphere $ \mathbb S ^{n}$ in dimension $ n+1$ with sparsity parameter $ 0 < s < n+1$ and assume that $ 0 < \delta < 1$. We show that for $ m \gtrsim \delta ^{-2} s \log \frac ns$, the one-bit map $$ x \mapsto \bigl[ sgn \langle x,g_j \rangle \bigr] _j=1 ^m, $$ where $ g_j$ are iid gaussian vectors on $ \mathbb R ^{n+1}$, with high probability has $ \delta $-RIP from $ K_s$ into the $ m$-dimensional Hamming cube. These bounds match the bounds for the linear $ \delta $-RIP given by $ x \mapsto \frac 1m[\langle x,g_j \rangle ] _{j=1} ^{m} $, from the sparse vectors in $ \mathbb R ^{n}$ into $ \ell ^{1}$. In other words, the one bit and linear RIPs are equally effective. There are corresponding improvements for other one-bit properties, such as the sign-product RIP property.
- A sign-linear one bit map from the $ d$-dimensional sphere $ \mathbb S ^{d}$ to the $ n$-dimensional Hamming cube $ H^n= \{ -1, +1\} ^{n}$ is given by $$ x \to { \mboxsign (x ⋅z_j) \;:\; 1≤j ≤n} $$ where $ \{z_j\} \subset \mathbb S ^{d}$. For $ 0 < \delta < 1$, we estimate $ N (d, \delta )$, the smallest integer $ n$ so that there is a sign-linear map which has the $ \delta $-restricted isometric property, where we impose normalized geodesic distance on $ \mathbb S ^{d}$, and Hamming metric on $ H^n$. Up to a polylogarithmic factor, $ N (d, \delta ) \approx \delta^{-2 + \frac2{d+1}}$, which has a dimensional correction in the power of $ \delta $. This is a question that arises from the one bit sensing literature, and the method of proof follows from geometric discrepancy theory. We also obtain an analogue of the Stolarsky invariance principle for this situation, which implies that minimizing the $L^2$ average of the embedding error is equivalent to minimizing the discrete energy $\sum_{i,j} \big( \frac12 - d(z_i,z_j) \big)^2$, where $d$ is the normalized geodesic distance.
- Nov 24 2015 math.CA arXiv:1511.07326v1In the current paper we present a new proof of the small ball inequality in two dimensions. More importantly, this new argument, based on an approach inspired by lacunary Fourier series, reveals the first formal connection between this inequality and discrepancy theory, namely the construction of two-dimensional binary nets, i.e. finite sets which are perfectly distributed with respect to dyadic rectangles. This relation allows one to generate all possible point distributions of this type. In addition, we outline a potential approach to the higher-dimensional small ball inequality by a dimension reduction argument. In particular this gives yet another proof of the two-dimensional signed (i.e. coefficients $\pm 1$) small ball inequality by reducing it to a simple one-dimensional estimate. However, we show that an analogous estimate fails to hold for arbitrary coefficients.
- In the current paper we obtain discrepancy estimates in exponential Orlicz and BMO spaces in arbitrary dimension $d \ge 3$. In particular, we use dyadic harmonic analysis to prove that for the so-called digital nets of order $2$ the BMO${}^d$ and $\exp \big( L^{2/(d-1)} \big)$ norms of the discrepancy function are bounded above by $(\log N)^{\frac{d-1}{2}}$. The latter bound has been recently conjectured in several papers and is consistent with the best known low-discrepancy constructions. Such estimates play an important role as an intermediate step between the well-understood $L_p$ bounds and the notorious open problem of finding the precise $L_\infty$ asymptotics of the discrepancy function in higher dimensions, which is still elusive.
- Jun 10 2013 math.NT arXiv:1306.1761v2It is a well-known conjecture in the theory of irregularities of distribution that the L1 norm of the discrepancy function of an N-point set satisfies the same asymptotic lower bounds as its L^2 norm. In dimension d=2 this fact has been established by Halasz, while in higher dimensions the problem is wide open. In this note, we establish a series of dichotomy-type results which state that if the L^1 norm of the discrepancy function is too small (smaller than the conjectural bound), then the discrepancy function has to be large in some other function space.
- We prove that in all dimensions n at least 3, for every integer N there exists a distribution of points of cardinality $ N$, for which the associated discrepancy function D_N satisfies the estimate an estimate, of sharp growth rate in N, in the exponential Orlicz class exp)L^2/(n+1). This has recently been proved by M.~Skriganov, using random digit shifts of binary digital nets, building upon the remarkable examples of W.L.~Chen and M.~Skriganov. Our approach, developed independently, complements that of Skriganov.
- A great challenge in the analysis of the discrepancy function D_N is to obtain universal lower bounds on the L-infty norm of D_N in dimensions d ≥3. It follows from the average case bound of Klaus Roth that the L-infty norm of D_N is at least (log N) ^(d-1)/2. It is conjectured that the L-infty bound is significantly larger, but the only definitive result is that of Wolfgang Schmidt in dimension d=2. Partial improvements of the Roth exponent (d-1)/2 in higher dimensions have been established by the authors and Armen Vagharshakyan. We survey these results, the underlying methods, and some of their connections to other subjects in probability, approximation theory, and analysis.
- In the present paper, we study the geometric discrepancy with respect to families of rotated rectangles. The well-known extremal cases are the axis-parallel rectangles (logarithmic discrepancy) and rectangles rotated in all possible directions (polynomial discrepancy). We study several intermediate situations: lacunary sequences of directions, lacunary sets of finite order, and sets with small Minkowski dimension. In each of these cases, extensions of a lemma due to Davenport allow us to construct appropriate rotations of the integer lattice which yield small discrepancy.
- The Small Ball Inequality is a conjectural lower bound on sums the L-infinity norm of sums of Haar functions supported on dyadic rectangles of a fixed volume in the unit cube. The conjecture is fundamental to questions in discrepancy theory, approximation theory and probability theory. In this article, we concentrate on a special case of the conjecture, and give the best known lower bound in dimension 3, using a conditional expectation argument.
- In [13], K. Roth showed that the expected value of the $L^2$ discrepancy of the cyclic shifts of the $N$ point van der Corput set is bounded by a constant multiple of $\sqrt{\log N}$, thus guaranteeing the existence of a shift with asymptotically minimal $L^2$ discrepancy, [11]. In the present paper, we construct a specific example of such a shift.
- 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.
- Mar 07 2008 math.CA arXiv:0803.0788v2When is the composition of paraproducts bounded? This is an important, and difficult question, related to to a question of Sarason on composition of Hankel matrices, and the two-weight problem for the Hilbert transform. We consider randomized variants of this question, finding non-classical characterizations, for dyadic paraproducts.
- Sep 18 2007 math.CA arXiv:0709.2713v1This paper is a companion to our prior paper arXiv:0705.4619 on the `Small Ball Inequality in All Dimensions.' In it, we address a more restrictive inequality, and obtain a non-trivial, explicit bound, using a single essential estimate from our prior paper. The prior bound was not explicit and much more involved.
- Let h_R denote an L ^∞ normalized Haar function adapted to a dyadic rectangle R contained in the unit cube in dimension d. We establish a non-trivial lower bound on the L^∞ norm of the `hyperbolic' sums $$ ∑_|R|=2 ^-n \alpha(R) h_R (x) $$ The lower bound is non-trivial in that we improve the average case bound by n^\eta for some positive \eta, a function of dimension d. As far as the authors know, this is the first result of this type in dimension 4 and higher. This question is related to Conjectures in (1) Irregularity of Distributions, (2) Approximation Theory and (3) Probability Theory. The method of proof of this paper gives new results on these conjectures in all dimensions 4 and higher. This paper builds upon prior work of Jozef Beck, from 1989, and first two authors from 2006. These results were of the same nature, but only in dimension 3.
- Sep 29 2006 math.CA arXiv:math/0609815v2We prove an inequality related to questions in Approximation Theory, Probability Theory, and to Irregularities of Distribution. Let $h_R$ denote an $L ^{\infty}$ normalized Haar function adapted to a dyadic rectangle $R\subset [0,1] ^{3}$. We show that there is a postive $\eta$ so that for all integers $n$, and coefficients $ \alpha (R)$ we have 2 ^-n \sum_\absR=2 ^-n \abs\alpha(R) ≲ n ^1 - \eta \NOrm \sum_\absR=2 ^-n \alpha(R) h_R >.∞. This is an improvement over the `trivial' estimate by an amount of $n ^{- \eta}$, and the optimal value of $\eta$ (which we do not prove) would be $ \eta =\frac12$. There is a corresponding lower bound on the $L ^{\infty}$ norm of the Discrepancy function of an arbitary distribution of a finite number of points in the unit cube in three dimensions. The prior result, in dimension 3, is that of József Beck \citeMR1032337, in which the improvement over the trivial estimate was logarithmic in $n$. We find several simplifications and extensions of Beck's argument to prove the result above.