Let $S$ be a dense subsemigroup of $(0,+infty)$. In this paper, we state   definition of thick near zero, and also  we will introduce a definition that is equivalent to the definition of piecewise syndetic near zero which presented by Hindman  and Leader in [6].  We define density near zero for subsets of $S$ by a collection of nonempty finite subsets of $S$ and we investigate the conditions un...

M. Beiglböck, V. Bergelson, and A. Fish proved that if G is a countable amenable group and A and B are subsets of G with positive Banach density, then the product set AB is piecewise syndetic. This means that there is a finite subset E of G such that EAB is thick, that is, EAB contains translates of any finite subset of G. When G = Z, this was first proven by R. Jin. We prove a quantitative ver...

We show that the constant colorings and the one-to-one colorings are insufficient for a canonical version of a certain theorem in Ramsey theory. Key phrases: van der Waerden’s theorem, arithmetic progression, piecewise syndetic, Ramsey Theory

We investigate when the set of finite products of distinct terms of a sequence 〈xn〉n=1 in a semigroup (S, ·) is large in any of several standard notions of largeness. These include piecewise syndetic, central , syndetic, central* , and IP*. In the case of a “nice” sequence in (S, ·) = (N,+) one has that FS(〈xn〉n=1) has any or all of the first three properties if and only if {xn+1 − ∑n t=1 xt : ...

Previous research extending over a few decades has established that multiplicatively large sets (in any of several interpretations) must have substantial additive structure. We investigate here the question of how much multiplicative structure can be found in additively large sets. For example, we show that any translate of a set of finite sums from an infinite sequence must contain all of the ...

There are several notions of largeness in a semigroup S that originated in topological dynamics. Among these are thick , central , syndetic and piecewise syndetic. Of these, central sets are especially interesting because they are partition regular and are guaranteed to contain substantial combinatorial structure. It is known that in (N,+) any central set may be partitioned into infinitely many...

We define and undertake a systematic study of thick, syndetic, and piecewise syndetic subsets of a Fräıssé structure. Each of these collections forms a family in the sense of Akin and Glasner [AG], and we define and study ultrafilters on each of these families, paying special attention to ultrafilters on the thick sets. In the process, we generalize many results of Bergelson, Hindman, and McCut...

In a semigroup, the combinatorial definitions of syndetic, piecewise syndetic, and IP are equivalent to their algebraic characterizations in terms of βS. We introduce the analogous definitions and characterizations of syndetic, piecewise syndetic, and IP for an adequate partial semigroup and show that equivalence between the combinatorial definition and algebraic characterization is lost once w...

In this work we prove that in the semigroup (N, +) if 〈xn〉n=1 is a sequence such that FS(〈xn〉n=1) is piecewise syndetic, then for any central* set A there exists a sum subsystem 〈yn〉n=1 of 〈xn〉n=1 with the property that FS(〈yn〉n=1) ∪ FP (〈yn〉n=1) ⊆ A.

An IP* set in a semigroup is one which must intersect the set of finite products from any specified sequence. (If the semigroup is noncommutative, one must specify the order of the products, resulting in “left” and “right” IP* sets.) If A is a subset of N with positive upper density, then the difference set A−A = {x ∈ N : there exists y ∈ A with x+ y ∈ A} is an IP* set in (N,+). Defining analog...