We prove that for any countable set A of real numbers, the set of binary indefinite quadratic forms Q such that the closure of Q ` Z2 ́ is disjoint from A has full Hausdorff dimension. Dedicated to S.G. Dani on the occasion of his 65th birthday

We characterize all compact and Hausdorff spaces X which satisfy that for every multiplicative bijection φ on C(X, I), there exist a homeomorphism μ : X −→ X and a continuous map p : X −→ (0,+∞) such that φ(f)(x) = f(μ(x)) for every f ∈ C(X, I) and x ∈ X. This allows us to disprove a conjecture of Marovt (Proc. Amer. Math. Soc. 134 (2006), 1065-1075). Some related results on other semigroups of...

The cardinal functions of pseudocharacter, closed pseudocharacter, and character are used to examine H-closed spaces and to contrast the differences between Hclosed and minimal Hausdorff spaces. An H-closed space X is produced with the properties that |X| > 22 ψ(X) and ψ(X) > 2ψ(X).

We study coupled random walks in the plane such that, at each step, the walks change direction by a uniform random angle plus an extra deterministic angle θ. We compute the Hausdorff dimension of the θ for which the walk has an unusual behavior. This model is related to a study of the spectral measure of some random matrices. The same techniques allow to study the boundary behavior of some Gaus...

Let R be a quasi-order on a compact Hausdorff topological space X. We prove that if X is scattered, then R satisfies the Priestley separation axiom if and only if R is closed in the product space X × X. Furthermore, if X is not scattered, then we show that there is a quasi-order on X that is closed in X × X but does not satisfy the Priestley separation axiom. As a result, we obtain a new charac...

Intrinsic Lp metrics are defined and shown to satisfy a dimension–free bound with respect to the Hausdorff metric. MSC 2000: 52A20, 52A27, 52A40, 60G15.

If X is a complete metric space, the collection of all non-empty compact subsets of X forms a complete metric space (H(X), h), where h is the Hausdorff metric. In this paper we explore some of the geometry of the space H(Rn). Specifically, we concentrate on understanding lines in H(R). In particular, we show that for any two points A, B ∈ H(Rn), there exist infinitely many points on the line jo...

In this paper, given two real space algebraic curves, not necessarily bounded, whose Hausdorff distance is finite, we provide bounds of their distance. These bounds are related to the distance between the projections of the space curves onto a plane (say, z = 0), and the distance between the z-coordinates of points in the original curves. Using these bounds we provide an estimation method for a...

We use the arithmetic-combinatorial method of Katz and Tao to give mixed-norm estimates for the x-ray transform on R when d ≥ 4. As an application, we obtain an improved estimate for the Hausdorff dimension of (d, k) sets, which are subsets of R containing a translate of every k-plane.