Let AN be an N-point set in the unit square and consider the discrepancy function

where x = (x1, x2) ∈ [0,1;]2, 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

The case of α = ∞ is the Theorem of Schmidt. This estimate is sharp. For the digit-scrambled van der Corput sequence, we have

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.

We consider an eigenvalue problem for a divergence-form elliptic operator Aε that has high-contrast periodic coefficients with period ε in each coordinate, where ε is a small parameter. The coefficients are perturbed on a bounded domain of “order one” size. The local perturbation of coefficients for such an operator could result in the emergence of localized waves—eigenfunctions whose corresponding eigenvalues lie in the gaps of the Floquet–Bloch spectrum. For the so-called double porosity-type scaling, we prove that the eigenfunctions decay exponentially at infinity, uniformly in ε Then, using the tools of twoscale convergence for high-contrast homogenization, we prove the strong two-scale compactness of the eigenfunctions of Aε. This implies that the eigenfunctions converge in the sense of strong two-scale convergence to the eigenfunctions of a two-scale limit homogenized operator A0, consequently establishing “asymptotic one-to-one correspondence” between the eigenvalues and the eigenfunctions of the operators Aε and A0. We also prove, by direct means, the stability of the essential spectrum of the homogenized operator with respect to local perturbation of its coefficients. This allows us to establish not only the strong two-scale resolvent convergence of Aε to A0 but also the Hausdorff convergence of the spectra of Aε to the spectrum of A0, preserving the multiplicity of the isolated eigenvalues.

For each integer n ≥ 2, let β(n) stand for the product of the exponents in the prime factorization of n. Given an arbitrary integer k ≥ 2, let nk be the smallest positive integer n such that β(n + 1) = β(n + 2) = … = β(n + k). We prove that there exist positive constants c1 and c2 such that, for all integers k ≥ 2,

We extend Kemperman's structure theorem by completely characterizing those finite subsets A and B of an arbitrary abelian group with |A + B| = |A| + |B|.

A group is said to be, finitely co-Hopfian when it contains no proper subgroup of finite index isomorphic to itself. It is known that irreducible lattices in semisimple Lie groups are finitely co-Hopfian. However, it is not clear, and does not appear to be known, whether this property is preserved under direct product. We consider a strengthening of the finite co-Hopfian condition, namely the existence of a non-zero multiplicative invariant, and show that, under mild restrictions, this property is closed with respect to finite direct products. Since it is also closed with respect to commensurability, it follows that lattices in linear semisimple groups of general type are finitely co-Hopfian.

In the kernel clustering problem we are given a large n × n positive-semidefinite matrix A = (aij) with and a small k × k positive-semidefinite matrix B = (bij). The goal is to find a partition S1, …, Sk of {1, … n} which maximizes the quantity

We study the computational complexity of this generic clustering problem which originates in the theory of machine learning. We design a constant factor polynomial time approximation algorithm for this problem, answering a question posed by Song et al. In some cases we manage to compute the sharp approximation threshold for this problem assuming the unique games conjecture (UGC). In particular, when B is the 3 × 3 identity matrix the UGC hardness threshold of this problem is exactly 16π/27. We present and study a geometric conjecture of independent interest which we show would imply that the UGC threshold when B is the k × k identity matrix is (8π/9)(1 – 1/k) for every k ≥ 3.

For a natural number n, let λ(n) denote the order of the largest cyclic subgroup of (ℤ/nℤ)*. For a given integer a, let Na(x) denote the number of n ≤ x coprime to a for which a has order λ(n) in (ℤ/nℤ)*. Let R(n) denote the number of elements of (ℤ/nℤ)* with order λ(n). It is natural to compare Na(x) with ∑n≤xR(n)/n. In this paper we show that the average of Na(x) for 1 ≤ a ≤ y is indeed asymptotic to this sum, provided y ≥ exp((2 + ε)(log x log log x)1/2), thus improving a theorem of the first author who had this for y ≥ exp((log x)3/4;). The result is to be compared with a similar theorem of Stephens who considered the case of prime numbers n.

We prove that Alexandrov spaces of non-negative curvature have Markov type 2 in the sense of Ball. As a corollary, any Lipschitz continuous map from a subset of an Alexandrov space of non-negative curvature into a 2-uniformly convex Banach space can be extended to a Lipschitz continuous map on the entire space.

Let υ be a valuation on K with value group Gυ, residue field kυ, rank υ = t and K (x1, …, xn) be the field of rational functions over K with n variables. If G is the direct sum of G1 and d infinite cyclic groups where G1 is a totally ordered group containing Gυ as an ordered subgroup with [G1 : Gυ] < ∞ and k′ is a finite field extension of kυ then there exists a residual transcendental extension u of υ to K (x1, …, xn) such that rank u = t + d, Gu = G the algebraic closure of kυ in kυ is k′ and trans deg ku/kυ = n − d.