Related papers: Arithmetic Progressions in Abundance by Combinator…

Furstenberg and Glasner proved that for an arbitrary k in N, any piecewise syndetic set contains k term arithmetic progression and such collection is also piecewise syndetic in Z. They used algebraic structure of beta N. The above result…

Furstenberg and Glasner proved that for an arbitrary k in N, any piecewise syndetic set contains k term arithmetic progressions and such collection is also piecewise syndetic in Z: They used algebraic structure of beta N. The above result…

Furstenberg and Glasner proved that for an arbitrary k in N, any piecewise syndetic set contains k length arithmetic progression and such collection is also piecewise syndetic in Z: They used algebraic structure of beta N. The above result…

H.Furstenberg and E.Glasner proved that for an arbitrary $k\in\mathbb{N}$, any piecewise syndetic set of integers contains a $k$-term arithmetic progression and the collection of such progressions is itself piecewise syndetic in…

Furstenberg, Glasscock, Bergelson, Beiglboeck have been studied abundance in arithmatic progression on various large sets like piecewise syndetic, central, thick, etc. but also there are so many sets in which abundance in progression is…

We present a self-contained proof of a strong version of van der Waerden's Theorem. By using translation invariant filters that are maximal with respect to inclusion, a simple inductive argument shows the existence of "piecewise…

Additive combinatorics is built around the famous theorem by Szemer\'edi which asserts existence of arithmetic progressions of any length among the integers. There exist several different proofs of the theorem based on very different…

The notion of abundance of certain type of configuration in certain large sets was first proved by Furstenberg and Glazner in 1998. After that many author investigate abundance of different types of configurations in different types of…

We give a purely combinatorial proof for a two-fold generalization of van der Waerden-Brauer's theorem and Hindman's theorem. We also give tower bounds for a finite version of it.

We answer a number of questions of Erd\H{o}s on the existence of arithmetic progressions in $k$-full numbers (i.e. integers with the property that every prime divisor necessarily occurs to at least the $k$-th power). Further, we deduce a…

The concept of Central sets, introduced by Furstenberg through the framework of topological dynamics, has played a pivotal role in combinatorial number theory. Furstenberg's Central Sets Theorem highlighted their rich combinatorial…

Let $AP_k=\{a,a+d,\ldots,a+(k-1)d\}$ be an arithmetic progression. For $\epsilon>0$ we call a set $AP_k(\epsilon)=\{x_0,\ldots,x_{k-1}\}$ an $\epsilon$-approximate arithmetic progression if for some $a$ and $d$, $|x_i-(a+id)|<\epsilon d$…

In this project we show the existence of arbitrary length arithmetic progressions in model sets and Meyer sets in the Euclidean $d$-space. We prove a van der Waerden type theorem for Meyer sets. We show that pure point subsets of Meyer sets…

B. Green and T. Tao have recently proved that 'the set of primes contains arbitrary long arithmetic progressions', answering to an old question with a remarkably simple formulation. The proof does not use any "transcendental" method and any…

H. Furstenberg introduced the notion of central set in terms of topological dynamics and established the central set theorem. The essence of central set theorem is that it is the simultaneous extension of van der Waerden's theorem and…

Let $G$ be a multiplicative subgroup of the prime field $\mathbb F_p$ of size $|G|> p^{1-\kappa}$ and $r$ an arbitrarily fixed positive integer. Assuming $\kappa=\kappa(r)>0$ and $p$ large enough, it is shown that any proportional subset…

In order to investigate multiplicative structures in additively large sets, Beiglb\"{o}ck et al. raised a significant open question as to whether or not every subset of the natural numbers with bounded gaps (syndetic set) contains…

The polynomial version of van der Waerden's theorem, proved using dynamical systems by V. Bergelson and A. Leibman in 1996, \cite{Bergelson1996}, significantly highlighted the role of dynamical systems in addressing problems related to…

The main motivation for this article is to explore the connections between the existence of certain combinatorial patterns (as in van der Corputs's theorem on arithmetic progressions of length $3$) with well-known tools and theorems for…

A famous theorem of Szemer\'edi asserts that given any density $0 < \delta \leq 1$ and any integer $k \geq 3$, any set of integers with density $\delta$ will contain infinitely many proper arithmetic progressions of length $k$. For general…