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We make the following three observations regarding a question popularized by Katznelson: is every subset of Z which is a set of Bohr recurrence also a set of topological recurrence? (i) If G is a countable abelian group and E ⊆ G is an I 0 set, then every subset of E - E which is a set of Bohr recurrence is also a set of topological recurrence. In particular every subset of { 2 n - 2 m : n , m ∈ N } which is a set of Bohr recurrence is a set of topological recurrence. (ii) Let Z ω be the direct sum of countably many copies of Z with standard basis E. If every subset of (E - E) - (E - E) which is a set of Bohr recurrence is also a set of topological recurrence, then every subset of every countable abelian group which is a set of Bohr recurrence is also a set of topological recurrence. (iii) Fix a prime p and let F p ω be the direct sum of countably many copies of Z / p Z with basis (e i) i ∈ N . If for every p-uniform hypergraph with vertex set N and edge set F having infinite chromatic number, the Cayley graph on F p ω determined by { ∑ i ∈ F e i : F ∈ F } has infinite chromatic number, then every subset of F p ω which is a set of Bohr recurrence is a set of topological recurrence. [ABSTRACT FROM AUTHOR]

We provide a necessary and sufficient condition to ensure that a multi-tile Ω ⊂ Rd of positive measure (but not necessarily bounded) admits a structured Riesz basis of exponentials for L2(Ω). New examples are given and this characterization is generalized to abstract locally compact abelian groups. [ABSTRACT FROM AUTHOR]

We consider certain finite sets of circle-valued functions defined on intervals of real numbers and estimate how large the intervals must be for the values of these functions to be uniformly distributed in an approximate way. This is used to establish some general conditions under which a random construction introduced by Katznelson for the integers yields sets that are dense in the Bohr group. We obtain in this way very sparse sets of real numbers (and of integers) on which two different almost periodic functions cannot agree, which makes them amenable to be used in sampling theorems for these functions. These sets can be made as sparse as to have zero asymptotic density or as to be t-sets, i.e., to be sets that intersect any of their translates in a bounded set. Many of these results are proved not only for almost periodic functions but also for classes of functions generated by more general complex exponential functions, including chirps or polynomial phase functions. [ABSTRACT FROM AUTHOR]

It is proved that if Ap is a countable elementary abelian p-group, then: (i) The ring End (Ap) does not admit a nondiscrete locally compact ring topology. (ii) Under (CH) the simple ring End (Ap)/I, where I is the ideal of End (Ap) consisting of all endomorphisms with finite images, does not admit a nondiscrete locally compact ring topology. (iii) The finite topology on End (Ap) is the only second metrizable ring topology on it. Moreover, a characterization of completely simple endomorphism rings of modules over commutative rings is obtained. [ABSTRACT FROM AUTHOR]

A letter to the editor is presented on the invariability of the positive limit set in a continuous discrete system.