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Sayan Goswami

Publications and source records attributed to Sayan Goswami.

At least 37 records · Page 2Linked to original sources
arXiv2024

Monochromatic Polynomial sumset structures on $\mathbb{N}$: an ultrafilter proof

Recently, using machinery's from Ergodic theory, Z. Lian, and R. Xiao proved if $P$ is any polynomial with no constant term, then for every finite coloring of $\mathbb{N}$, there exists two infinite subsets $B,C$ of $\mathbb{N}$ such that the set $\{P(b)+P(c):b\in B, c\in C\}$ is monochromatic. In this article we improve their result by proving that instead of taking such polynomials we can choose any function $f$ having the property that $f(\mathbb{N})\setminus \mathbb{N}$ is finite. We use ultrafilter techniques to prove our result.

math.CO↗
arXiv2023

Combined exponential patterns in multiplicative $IP^{\star}$ sets

$IP$ sets play fundamental role in arithmetic Ramsey theory. A set is called an additive $IP$ set if it is of the form $FS\left(\langle x_{n}\rangle_{n\in \mathbb{N}}\right)=\left\{ \sum_{t\in H}x_{t}:H\right.$ is a nonempty finite subset of $\left.\mathbb{N}\right\}$, whereas it is called a multiplicative $IP$ set if it is of the form $FP\left(\langle x_{n}\rangle_{n\in \mathbb{N}}\right)=\left\{ \prod_{t\in H}x_{t}:H\right.$ is a nonempty finite subset of $\left. \mathbb{N}\right\}$ for some injective sequence $\langle x_{n}\rangle_{n\in \mathbb{N}}.$ An additive $IP^{\star}$ (resp. multiplicative $IP^{\star}$) set is a set which intersects every additive $IP$ set (resp. multiplicative $IP$ set). In \cite{key-1}, V. Bergelson and N. Hindman studied how rich additive $IP^{\star}$ sets are. They proved additive $IP^{\star}$ sets ($AIP^{\star}$ in short) contain finite sums and finite products of a single sequence. An analogous study was made by A. Sisto in\cite{key-3}, where he proved that multiplicative $IP^{\star}$ sets ($MIP^{\star}$ in short) contain exponential tower\footnote{will be defined later} and finite product of a single sequence. However exponential patterns can be defined in two different ways. In this article we will prove that $MIP^{\star}$ sets contain two different exponential patterns and finite product of a single sequence. This immediately improves the result of A. Sisto. We also construct a $MIP^\star$ set, not arising from the recurrence of measurable dynamical systems. Throughout our work we will use the machinery of the algebra of the Stone-Čech Compactification of $\mathbb{N}$.

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arXiv2023

On the set of Kronecker numbers

An positive even number is said to be a Kronecker number if it can be written in infinitely many ways as the difference between two primes, and it is believed that all even numbers are Kronecker numbers. We will study the division and multiplication of Kronecker numbers to study the largeness of the set of Kronecker numbers. A numerical lower bound for the density Kronecker numbers among even numbers is given, and it is proved that there exists a computable constant $k$ and a set $D$ consisting of at most 720 computable Maillet numbers such that, for any integer $n$, $kn$ can be expressed as a product of a Kronecker number with a Maillet number in $D$. Meanwhile, it is proved that every positive rational number can be written as a ratio of two Kronecker numbers.

math.NT↗
arXiv2023

Product of difference sets of set of primes

In a recent work \cite{key-11}, A. Fish proved that if $E_{1}$ and $E_{2}$ are two subsets of $\mathbb{Z}$ of positive upper Banach density, then there exists $k\in\mathbb{Z}$ such that $k\cdot\mathbb{Z}\subset\left(E_{1}-E_{1}\right)\cdot\left(E_{2}-E_{2}\right).$ In this article we will show that a similar result is true for the set of primes $\mathbb{P}$ (which has density $0$). We will prove that there exists $k\in\mathbb{N}$ such that $k\cdot\mathbb{N}\subset\left(\mathbb{P}-\mathbb{P}\right)\cdot\left(\mathbb{P}-\mathbb{P}\right),$ where $\mathbb{P}-\mathbb{P}=\left\{ p-q:p>q\,\text{and}\,p,q\in\mathbb{P}\right\} .$

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arXiv2022

Some remarks on Alweiss's technique and monochromatic configuration of the form $\left\{ x,y,x+y,x\cdot y\right\} $ over Rationals

In this article, we will explore a recent method of Alweiss \cite{key-1} using ultrafilter technique to study monochromatic partition regular structure of the form $\left\{ x,y,x+y,x\cdot y\right\} $ over rationals, which is recently proved by Bowen, and Sabok in \cite{key-17}. Our methods explore that each member of combinatorially rich ultrafilters contains these types of configurations. Besides this, we will also prove that for any $n\in\mathbb{N},$ these sets will contain configuration of the form $\left\{ x,y,x+y,x\cdot y^{n}\right\}$, partially proved by Xiao in \cite{key-25}.

math.CO↗
arXiv2022

An improved threshold for the number of distinct intersections of intersecting families

A family $\mathcal{F}$ of subsets of $\{1,2,\ldots,n\}$ is called a $t$-intersecting family if $|F\cap G| \geq t$ for any two members $F, G \in \mathcal{F}$ and for some positive integer $t$. If $t=1$, then we call the family $\mathcal{F}$ to be intersecting. Define the set $\mathcal{I}(\mathcal{F}) = \{F\cap G: F, G \in \mathcal{F} \text{ and } F \neq G\}$ to be the collection of all distinct intersections of $\mathcal{F}$. Frankl et al. proved an upper bound for the size of $\mathcal{I}(\mathcal{F})$ of intersecting families $\mathcal{F}$ of $k$-subsets of $\{1,2,\ldots,n\}$. Their theorem holds for integers $n \geq 50 k^2$. In this article, we prove an upper bound for the size of $\mathcal{I}(\mathcal{F})$ of $t$-intersecting families $\mathcal{F}$, provided that $n$ exceeds a certain number $f(k,t)$. Along the way we also improve the threshold $k^2$ to $k^{3/2+o(1)}$ for the intersecting families.

math.CO↗
arXiv2022

A note on distinct differences in $t$-intersecting families

For a family $\mathcal{F}$ of subsets of $\{1,2,\ldots,n\}$, let $\mathcal{D}(\mathcal{F}) = \{F\setminus G: F, G \in \mathcal{F}\}$ be the collection of all (setwise) differences of $\mathcal{F}$. The family $\mathcal{F}$ is called a $t$-intersecting family, if for some positive integer $t$ and any two members $F, G \in \mathcal{F}$ we have $|F\cap G| \geq t$. The family $\mathcal{F}$ is simply called intersecting if $t=1$. Recently, Frankl proved an upper bound on the size of $\mathcal{D}(\mathcal{F})$ for the intersecting families $\mathcal{F}$. In this note we extend the result of Frankl to $t$-intersecting families.

math.CO↗
arXiv2022

Additive and multiplicative Gower's Ramsey theorem

W. T. Gower generalized Hindman's Finite sum theorem over $X_{k}=\left\{ \left(n_{1},n_{2},\ldots,n_{k}\right):n_{1}\neq0\right\} $ by showing that for any finite coloring of $X_{k}$ there exists a sequence such that the Gower subspace generated by that sequence is monochromatic. For $k=1,$ this immediately gives the finite sum theorem. In this article we will show that for any finite coloring of $X_{k}$ there exist two sequences $\left\{ \mathbf{n_{i}}:i\in I\right\} $ and $\left\{ \mathbf{m_{i}}:i\in I\right\} $ such that the Gower subspace generated by $\left\{ \mathbf{n_{i}}:i\in I\right\} $ and set of all finite products of $\left\{ \mathbf{m_{i}}:i\in I\right\} $ are in a single color. This immediately generalize a result of V. Bergelson and N. Hindman which says that for any finite coloring of $\mathbb{N}$, there exist two sequences $\left(x_{n}\right)_{n}$ and $\left(y_{n}\right)_{n}$ such that the finite sum and product generated by $\left(x_{n}\right)_{n}$ and $\left(y_{n}\right)_{n}$ are in a same color.

math.CO↗
arXiv2022

Polynomial extension of the Stronger Central Sets Theorem

In [F81] Furstenberg introduced the notion of central set and established his famous Central Sets Theorem. Since then, several improved versions of Furstenberg's result have been found. The strongest generalization has been published by De, Hindman and Strauss in [DHS08], whilst a polynomial extension by Bergelson, Johnson and Moreira appeared in [BJM17]. In this article, we will establish a polynomial extension of the stronger version of the central sets theorem, and we will discuss properties of the families of sets that this result leads to consider

math.CO↗
arXiv2022

An infinitary Zero sum theorem

Erdős-Ginzburg-Ziv theorem says that if there are 2n-1 number is given, then there are n numbers such that their sum is divided by n. We will connect this theorem with the Ramsey theoretic large sets and will prove an infinitary version of this theorem. In our proof we will use the methods of ultrafilters. But one may proceed using methods of Topological dynamics.

math.CO↗
arXiv2021

Hales-Jewett type configurations in small sets

In a recent work, N. Hindman, D. Strauss and L. Zamboni have shown that the Hales-Jewett theorem can be combined with a sufficiently well behaved homomorphisms. Their work was completely algebraic in nature, where they have used the algebra of Stone-Cech compactification of discrete semigroup. They have proved the existence of those configurations in piecewise syndetic sets, which is a Ramsey theoretic rich set. In our work we will show those forms are still present in very small but Ramsey theoretic sets, (like J-set, C-set) and our proof is purely elementary in nature.

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arXiv2021

Dynamical IP$^{\star}$-sets in weak rings

V. Bergelson and N. Hindman proved that IP$^{\star}$- sets contain all possible finite sums and products of a sum subsystem of any sequence in $\mathbb{N}$. In a recent work the second author of this article has proved that a stronger result holds for dynamical IP$^{\star}$- sets. In this article we will establish a non-commutative version of this result. We will prove that a richer configuration is contained in dynamical IP$^{\star}$- sets in weak rings.

math.CO↗
arXiv2021

Central Sets Theorem along filters and some combinatorial consequences

The Central Sets Theorem was introduced by H. Furstenberg and then afterwards several mathematicians have provided various versions and extensions of this theorem. All of these theorems deal with central sets, and its origin from the algebra of Stone-Cech compactification of arbitrary semigroup, say $βS$. It can be proved that every closed subsemigroup of $βS$ is generated by a filter. We will show that, under some restrictions, one can derive the Central Sets Theorem for any closed subsemigroup of $βS$ . We will derive this theorem using the corresponding filter and its algebra. Later we will also deal with how the notions of largeness along filters are preserved under some well behaved homomorphisms and give some consequences.

math.CO↗
arXiv2021

Elementary characterization of essential F-sets and its combinatorial consequences

There is a long history of studying Ramsey theory using the algebraic structure of the Stone-Čech compactification of discrete semigroup. It has been shown that various Ramsey theoretic structures are contained in different algebraic large sets. In this article we will deduce the combinatorial characterization of certain sets, that are the member of the idempotent ultrafilters of the closed subsemigroup of $βS$, arising from certain Ramsey family. In a special case when $S=\mathbb{N}$, we will deduce that sets which are the members of all idempotent ultrafilters of those semigroups contain certain additive-multiplicative structures. Later we will generalize this result for weak rings, where we will show a non-commutative version of the additive-multiplicative structure.

math.GN↗
arXiv2021

Some new symmetric structures in Ramsey theory

In this article, we will investigate several new configurations in Ramsey Theory, using the $\ostar_{l,k}$-operation on the set of integers, recently introduced in \cite{key-4}. This operation is useful to study symmetric structures in the set of integers, such as monochromatic configurations of the form $\left\{ x,y,x+y+xy\right\} $ as one of its simplest case. In \cite{key-4}, the author has studied more general symmetric structures. It has been shown that the Hindman's Theorem, van der Waerden's Theorem, Deuber's Theorem have their own symmetric versions. In this article we will explore several new structures, including polynomial versions of these symmetric structures and some of its variants. As a result, we get several new symmetric polynomial configurations as well as new linear symmetric patterns. In the final section, we will also introduce two new operations on the set of non-negative integers $\mathbb{N}$, to obtain further new configurations.

math.CO↗
arXiv2021

Dynamical characterization of central sets along filter

Using the notions of Topological dynamics, H. Furstenberg defined central sets and proved the Central Sets Theorem. Later V. Bergelson and N. Hindman characterized central sets in terms of algebra of the Stone-Čech Compactification of discrete semigroup. They found that central sets are the members of the minimal idempotents of $βS$, the Stone-Čech Compactification of a semigroup $\left(S,\cdot\right)$. We know that any closed subsemigroup of $βS$ is generated by a filter. We call a set $A$ to be a $\mathcal{F}$-central set if it is a member of a minimal idempotent of a closed subsemigroup of $βS$, generated by the filter $\mathcal{F}$. In this article we will characterize the $\mathcal{F}$-central sets dynamically.

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