The famous theorem of Cobham says that, for multiplicatively independent integers k and l, any subset of N, which is both kand l-recognizable, is recognizable. Many of its proofs are based on so called Hansel’s lemma stating that such a kand l-recognizable set is syndetic. We consider these proofs and point out that some of them are inadequate.

Let F 6= ∅ be a closed subset of Rn with empty interior. There are several proposals what should be called the dimension of F , globally and locally. Besides the classical Hausdorff dimension there exist nearby but, in general, not identical definitions, better adapted to the needs of measure theory, see [14] and also [9] and [7]: Ch.II,1. One aim of our paper is to contribute to this field of ...

Problem B1 (20 pts). A ray , R, of E is any subset of the form R = {a+ tu | t ∈ R, t ≥ 0}, for some point, a ∈ E and some nonzero vector, u ∈ R. A subset, A ⊆ E, is unbounded iff it is not contained in any ball. Prove that every closed and unbounded convex set, A ⊆ E, contains a ray. Hint : Pick some point, c ∈ A, as the origin and consider the sphere, S(c, r) ⊆ E, of center c and radius r > 0....

We have already considered instances of the following type of problem: given a bounded subset Ω of Euclidean space R N , to determine #(Ω ∩ Z N), the number of integral points in Ω. It is clear however that there is no answer to the problem in this level of generality: an arbitrary Ω can have any number of lattice points whatsoever, including none at all. In [Gauss's Circle Problem], we counted...

This is a survey of the results in [1, 2, 3, 4, 5, 6] on convexity numbers of closed sets in Rn, homogeneity numbers of continuous colorings on Polish spaces and families of functions covering Rn. 1. Convexity numbers of closed sets in R A natural way to measure the degree of non-convexity of a subset S of a real vector space is the convexity number γ(S), the least size of a family C of convex ...

We prove that a closed, geodesically convex subset C of P 2 (R) is closed with respect to weak convergence in P 2 (R). This means that if (μn) ⊂ C is such that μn ⇀ μ in duality with continuous bounded functions and supn ∫ |x|dμn <∞, then μ ∈ C as well.

A first-order expansion of the R-vector space structure on R does not define every compact subset of every Rn if and only if topological and Hausdorff dimension coincide on all closed definable sets. Equivalently, if A ⊆ Rk is closed and the Hausdorff dimension of A exceeds the topological dimension of A, then every compact subset of every Rn can be constructed from A using finitely many boolea...