It is known that if N is finitely colored, then some color class is piecewise syndetic. (See Definition 1.1 below for a definition of piecewise syndetic.) We generalize this result by considering finite colorings of the set of all finite subsets of N . The monochromatic objects obtained are “d-copies” of arbitrary finite forests and arbitrary infinite forests of finite height. Van der Waerden’s...

1. Basic properties of ultrafilters 2. Applications of ultrafilters. The space βS 3. Semigroups: basic facts and examples 4. βS as a semigroup 5. Hindman’s finite sums theorem 6. Semigroups: ideals and minimal idempotents 7. The Hales-Jewett theorem 8. Large sets: thick, central, IP-, and syndetic sets 9. Large sets: piecewise syndetic sets 10. Dynamical systems: basic notions 11. βS as a dynam...

Sets that enjoy with recurrence properties near zero, in particular, sets force contain broken IP-set uniform zero and syndetic set have been introduced some relations among them are established on countable dense subsemigroup of the additive semigroup positive real numbers. The article begins a characterization recurrent point affects results exhibited here. It is also showed how can be realiz...

It is well known and not difficult to prove that if C ⊆ Z has positive upper Banach density, the set of differences C −C is syndetic, i.e. the length of gaps is uniformly bounded. More surprisingly, Renling Jin showed that whenever A and B have positive upper Banach density, then A − B is piecewise syndetic. Jin’s result follows trivially from the first statement provided that B has large inter...

Answering a problem posed by Keisler and Leth, we prove a theorem in nonstandard analysis to reveal a phenomenon about sumsets, which says that if two sets A and B are large in terms of \measure", then the sum A+B is not small in terms of \order{topology". The theorem has several corollaries about sumset phenomenon in the standard world; these are described in sections 2{4. One of these is a ne...

Hindman and Leader introduced the set $$0^{+}$$ of ultrafilters on $$\left( 0,1\right) $$ , characterized smallest ideal 0^{+},+\right) proved Central Set Theorem near 0. Recently Polynomial has been by Bergelson, Johnson Jr. Moreira. In this note, we will

In this note, we give an alternative proof of the following result. Let p, q ≥ 2 be two multiplicatively independent integers. If an infinite set of integers is both pand q-recognizable, then it is syndetic. Notice that this result is needed in the classical proof of the celebrated Cobham’s theorem. Therefore the aim of this paper is to complete [13] and [1] to obtain an accessible proof of Cob...

For a sequence {Tn} of continuous linear operators on a separable Fréchet space X, we discuss necessary conditions and sufficient conditions for {Tn} to be topologically mixing, and the relations between topological mixing and the Hypercyclicity Criterion. Among them are: 1) topological mixing is equivalent to being hereditarily densely hypercyclic; 2) the Hypercyclicity Criterion with respect ...