arXiv subjects

Sayan Goswami

Publications and source records attributed to Sayan Goswami.

At least 19 recordsLinked to original sources

arXiv2026

A Uniform Product-Difference Theorem for Dense Subsets of $\mathbb Z^2$

We establish a uniform product-difference theorem for dense subsets of $\mathbb Z^2$, which gives an affirmative answer to Problem~2 of Fish and, as consequences, to both parts of his Problem~1. More precisely, we prove that for every $\delta>0$ there exists an integer $K(\delta)\geq 1$ such that every set $E\subseteq\mathbb Z^2$ with upper Banach density $d^\star(E)\geq\delta$ satisfies \[ K(\delta)\mathbb Z \subseteq \{ab:(a,b)\in E-E\}. \] As consequences, we obtain affirmative answers to both parts of Fish's Problem~1: for positive-density sets $E_1,E_2\subseteq\mathbb Z$ and $E\subseteq\mathbb Z^2$, respectively, the sets \[ (E_1-E_1)^2-(E_2-E_2)^2 \quad\text{and}\quad \{x^2-y^2:(x,y)\in E-E\} \] contain nontrivial ideals of $\mathbb Z$, with generators depending only on the corresponding density thresholds. In particular, the latter result also settles a conjecture of Davies concerning differences of the indefinite quadratic form $x^2-y^2$ in dense subsets of $\mathbb Z^2$.

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arXiv2026

On difference sets of dense subsets of $\mathbb{Z}^2$

In this article, we study the structure of the difference set $E - E$ for subsets $E \subseteq \mathbb{Z}^2$ of positive upper Banach density. Fish asked in [Proc. Amer. Math. Soc. 146 (2018), 3449-3453] whether, for every such set $E$, there exists a nonzero integer $k$ such that $k \cdot \mathbb{Z} \subseteq \{\, xy : (x,y) \in E - E \,\}.$ Although this question remains open, we establish a relatively weaker form of this conjecture. Specifically, we prove that if $\langle a_j\rangle_{j=1}^m$ is any finite sequence in $\mathbb{N},$ then there exist infinitely many integers $k \in \mathbb{Z}$ and a sequence $\langle x_n \rangle_{n \in \mathbb{N}}$ in $\mathbb{Z}$ such that $k \cdot MT\left(\langle a_j \rangle_{j=1}^m, \langle x_n\rangle_{n}\right) \subseteq \{\, xy : (x,y) \in E - E \,\},$ where $MT\left(\langle a_j \rangle_{j=1}^m, \langle x_n\rangle_{n}\right)$ denotes the milliken-Taylor configuration generated by the sequences $\langle a_j\rangle_{j=1}^m$ and $\langle x_n \rangle_{n \in \mathbb{N}}$.

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arXiv2025

Random Finite Sumsets and Product Sets in Subsets of the Natural Numbers

We investigate the occurrence of additive and multiplicative structures in random subsets of the natural numbers. Specifically, for a Bernoulli random subset of $\mathbb{N}$ where each integer is included independently with probability $p\in (0,1)$, we prove that almost surely such a set contains finite sumsets (FS-sets) and finite product sets (FP-sets) of every finite length. In addition, we establish a novel connection between Hindman's partition theorem and the central limit theorem, providing a probabilistic perspective on the asymptotic Gaussian behavior of monochromatic finite sums and products. These results can be interpreted as probabilistic analogues of finite-dimensional versions of Hindman's theorem. Applications, implications, and open questions related to infinite FS-sets and FP-sets are discussed.

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arXiv2025

The Interplay between Additive and Multiplicative Central Sets Theorems

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 structure. Later, De, Hindman, and Strauss strengthen this theorem using the algebraic framework of the Stone--\v{C}ech compactification. In this article, we establish a unified version of the Central Sets Theorem that simultaneously captures both additive and multiplicative structures.

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arXiv2025

Matrix Formulation of Moreira Theorem

In a celebrated article, Moreira proved for every finite coloring of the set of naturals, there exists a monochromatic copy of the form $\{x,x+y,xy\},$ which gives a partial answer to one of the central open problems of Ramsey theory asking whether $\{x,y,x+y,xy\}$ is partition regular. In this article, we prove the matrix version of the Moreira theorem. We prove that if $A$ and $B$ are two finite image partition regular matrices of the same order, then for every finite coloring of the set of naturals, there exist two vectors $\overrightarrow{X}, \overrightarrow{Y}$ such that $\{A\overrightarrow{X}, A\overrightarrow{X}+B\overrightarrow{Y}, A \overrightarrow{X}\cdot B\overrightarrow{Y}\}$ is monochromatic, where addition and multiplication are defined coordinate-wise.

math.CO↗

arXiv2025

Homogeneous Patterns in Ramsey Theory

In this article, we investigate homogeneous versions of certain nonlinear Ramsey-theoretic results, with three significant applications. As the first application, we prove that for every finite coloring of $\mathbb{Z}^+$, there exist an infinite set $A$ and an arbitrarily large finite set $B$ such that $A \cup (A+B) \cup A \cdot B$ is monochromatic. This result resolves the finitary version of a question posed by Kra, Moreira, Richter, and Robertson regarding the partition regularity of $(A+B) \cup A \cdot B$ for infinite sets $A, B$ (see (Question 8.4, J. Amer. Math. Soc., 37 (2024))), which is closely related to a question of Erd\H{o}s. As the second application, we make progress on a nonlinear extension of the partition regularity of Pythagorean triples. Specifically, we demonstrate that the equation $x^2 + y^2 = z^2 + P(u_1, \dots, u_n)$ is $2$-regular for certain appropriately chosen polynomials $P$ of any desired degree. Finally, as the third application, we establish a nonlinear variant of Rado's conjecture concerning the degree of regularity. We prove that for every $m, n \in \mathbb{Z}^+$, there exists an $m$-degree homogeneous equation that is $n$-regular but not $(n+1)$-regular. The case $m = 1$ corresponds to Rado's conjecture, originally proven by Alexeev and Tsimerman (J. Combin. Theory Ser. A, 117 (2010), and later independently by Golowich (Electron. J. Combin. 21 (2014)).

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arXiv2024

Monochromatic Translated Product and Answering Sahasrabudhe's Conjecture

This article resolves two related problems in Ramsey theory on the integers. We show that for any finite coloring of the set of natural numbers, there exist numbers $a$ and $b$ for which the configuration $\{a, b, ab, a(b+1)\}$ is monochromatic. By redefining the variables $a=x$ and $ab=y,$ our configurations transforms into $\{x,y,x+y,\frac{y}{x}\}.$ This finding has two main consequences: first, it disproves a conjecture proposed by J. Sahasrabudhe; second, it establishes a quotient version of the long-standing Hindman's conjecture, which asks for a monochromatic set of the form $\{x,y,x+y,xy\}$.

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arXiv2024

Exponential Schur and Hindman Theorem in Ramsey Theory

Answering a conjecture of A. Sisto, J. Sahasrabudhe proved the exponential version of the Schur theorem: for every finite coloring of the naturals, there exists a monochromatic copy of $\{x,y,x^y:x\neq y\},$ which initiates the study of exponential Ramsey theory in arithmetic combinatorics. In this article, We first give two short proofs of the exponential Schur theorem, one using Zorn's lemma and another using $IP_r^\star$ van der Waerden's theorem. Then using the polynomial van der Waerden theorem iteratively we give a proof of the exponential Hindman theorem. Then applying our results we prove for every natural number $m,n$ the equation $x_n^{x_{n-1}^{\cdot^{\cdot^{\cdot^{x_1}}}}}=y_1\cdots y_m$ is partition regular, which can be considered as the exponential version of a more general version of the P. Csikv\'{a}ri, K. Gyarmati, and A. S\'{a}rk\"{o}zy conjecture, which was solved by V. Bergelson and N. Hindman independently. As a consequence of our results, we also prove that for every finite partition of $\mathbb{N},$ there exists two different sequences $\langle x_n\rangle_n$ and $\langle y_n\rangle_n$ such that both the multiplicative and exponential version of Hindman theorem generated by these sequences resp. are monochromatic, whereas in the counterpart in the finitary case, J. Sahasrabudhe proved that both sequences are the same. Our result can be considered as an exponential analog to the result of V. Bergelson and N. Hindman. We also prove that a large class of ultrafilters with certain properties do not exist, which could give us direct proof of the exponential Schur theorem. This result can be thought of as partial evidence of the nonexistence of Galvin-Glazer's proof of the exponential Hindman theorem.

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arXiv2024

On dynamical $C^{\star}$-set and its combinatorial consequences

Using the methods from topological dynamics, H. Furstenberg introduced the notion of a central set and proved the famous Central Sets Theorem. Later D. De, Neil Hindman, and D. Strauss [Fund. Math.199 (2008), 155-175.] established a stronger version of the Central Sets Theorem and then introduced the notion of $C$-sets satisfying the Central Sets Theorem and studied the properties of these sets. For any weak mixing system $\left(X, \mathcal{B},\mu, T\right),$ and $A_{0},A_{1}\in\mathcal{B}$, with $\mu\left(A_{0}\right)\mu\left(A_{1}\right)>0$, R. Kung and X.Ye [Disc. Cont. Dyn. sys., 18 (2007) 817-827.] proved that the set $N\left(A,B\right)= \left\{n:\mu\left(A_{0}\cap T^{-n}A_{1}\right)>0\right\}$ intersects all sets of positive upper Banach density. However, later N. Hindman and D. Strauss [New York J. Math. 26 (2020) 230-260.] proved that there exist $C$-sets having zero upper Banach density. Inspired by this result, in this article, we prove that $N\left(A, B \right)$ intersects with all $C$-sets. Then we introduce the notion of a dynamical $C^{\star}$-set and then we study their combinatorial properties.

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arXiv2024

Rawsamble: Overlapping and Assembling Raw Nanopore Signals using a Hash-based Seeding Mechanism

Raw nanopore signal analysis is a common approach in genomics to provide fast and resource-efficient analysis without translating the signals to bases (i.e., without basecalling). However, existing solutions cannot interpret raw signals directly if a reference genome is unknown due to a lack of accurate mechanisms to handle increased noise in pairwise raw signal comparison. Our goal is to enable the direct analysis of raw signals without a reference genome. To this end, we propose Rawsamble, the first mechanism that can identify regions of similarity between all raw signal pairs, known as all-vs-all overlapping, using a hash-based search mechanism. We use these overlaps to construct de novo assembly graphs with an existing assembler, miniasm, off-the-shelf. To our knowledge, these are the first de novo assemblies ever constructed directly from raw signals without basecalling. Our extensive evaluations across multiple genomes of varying sizes show that Rawsamble provides a significant speedup (on average by 5.01x and up to 23.10x) and reduces peak memory usage (on average by 5.74x and up to by 22.00x) compared to a conventional genome assembly pipeline using the state-of-the-art tools for basecalling (Dorado's fastest mode) and overlapping (minimap2) on a CPU.We find that around one-third of Rawsamble 's overlapping pairs are also found by minimap2. We find that when we use overlapping reads from Rawsamble, we can construct unitigs that are 1) as accurate as those built from minimap2's overlaps and 2) up to half a chromosome in length (e.g., 2.3 million bases for E. coli). Source code: https://github.com/CMU-SAFARI/RawHash

q-bio.GN↗

arXiv2024

Restricted van der Waerden theorem for nilprogressions

In [Adv. Math., 321 (2017) 269-286], using the theory of ultrafilters, J. H. Johnson Jr., and F. K. Richter proved the nilpotent polynomial Hales-Jewett theorem. Using this result they proved the restricted version of the van der Waerden theorem for nilprogressions of rank $2$ and conjectured that this result must hold for arbitrary rank. In this article, we give an affirmative answer to their conjecture.

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arXiv2024

Nonlinear Kernel Partition Regularity: Necessary and Sufficient Conditions

A matrix $ A $ is called \emph{kernel partition regular} if, for every finite coloring of the natural numbers $ \mathbb{N} $, there exists a monochromatic solution to the equation $ A\vec{X} = 0 $. In 1933, Rado characterized such matrices by showing that a matrix is kernel partition regular if and only if it satisfies the so-called \emph{column condition}. In this article, we investigate polynomial extensions of Rado's theorem by studying systems of nonlinear equations of the form $A \vec{X} + P(z) = \vec{0},$ where $A$ is a matrix with integer entries and $P$ is a finite set of polynomials in one variable with no constant term. We present several nonlinear systems of equations that are kernel partition regular, showing that the classical column condition still guarantees kernel partition regularity, even when the system is extended by adding a nonlinear polynomial term. We then establish a structural necessary condition for the partition regularity of nonlinear Rado-type systems, extending the classical column condition to a nonlinear setting. This condition generalizes Rado's classical column condition by exploring the dependencies between the linear and higher-degree polynomial components of the system.

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arXiv2024

Dynamical characterization of central sets in adequate partial semigroups

Using the methods from topological dynamics, H. Furstenberg introduced the notions of Central sets and proved the famous Central Sets Theorem which is the simultaneous extension of the van der Waerden and Hindman Theorem. Later N. Hindman and V. Bergelson found an equivalent formulation of Central sets in the set of natural numbers in terms of the algebra of the Stone-\v{C}ech compactification of discrete semigroups. The general case was proved by H. Shi and H. Yang. Using the notions of ultrafilters, J. McLeod introduced the notions of Central sets for commutative adequate partial semigroups, however for noncommutative cases, Central sets can be defined similarly. In this article, introducing the notions of topological dynamics for partial semigroup actions, we find an equivalent dynamical characterization of central sets in partial semigroups\footnote{Recently in \cite{GTG}, authors attempted to do the same but in a different approach.}. Throughout our article, we follow the approach of N. Hindman and D. Strauss [N. Hindman, and D. Strauss: Algebra in the Stone-\v Cech compactification: theory and applications, second edition, de Gruyter, Berlin, 2012.].

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arXiv2024

An Explicit Ramsey Bound for de Polignac Numbers

A number $ m $ is called a \textbf{de Polignac number}($ \mathbf{POL} $ in short) if it can be expressed as the difference of infinitely many pairs of consecutive prime numbers. For any given $r\in \mathbb N,$ a set $A$ is said to be $\Delta_r^\star$ if for all sets $S$ with $|S|=r$ such that $A\cap \{s-t:s>t\in S\}\neq \emptyset.$ In this article we prove that the set $\mathbf{POL}$ is $\Delta_r^\star$ with the specific computable value of $r,$ where $r=exp(\mathcal{O}(50)).$ Then we prove that there exists a set $E$ with $|E|\leq 2^{50\cdot \prod_{i=1}^{49}p_i} $ (where $(p_i)_i$ is the enumeration of primes) such that $\mathbb{N}=\bigcup_{t\in E} t^{-1}\mathbf{POL}.$

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arXiv2024

Multidimensional Stronger Central Sets Theorem and its Polynomial Extension

We establish and fully characterize the multidimensional extension of the Stronger Central Sets Theorem. Additionally, we develop a polynomial generalization of this result. Our approach utilizes tools from the Algebra of the Stone-\v{C}ech compactification of discrete semigroups. Several applications of these results are also discussed.

math.CO↗

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↗

arXiv2024

The interplay between additive and symmetric large sets and their combinatorial applications

The study of symmetric structures is a new trend in Ramsey theory. Recently in [7], Di Nasso initiated a systematic study of symmetrization of classical Ramsey theoretical results, and proved a symmetric version of several Ramsey theoretic results. In this paper Di Nasso asked if his method could be adapted to find new non-linear Diophantine equations that are partition regular [7,Final remarks (4)]. By analyzing additive, multiplicative, and symmetric large sets, we construct new partition regular equations that give a first affirmative answer to this question. A special case of our result shows that if $P$ is a polynomial with no constant term then the equation $x+P(y-x)=z+w+zw$, where $y\neq x$ is partition regular. Also we prove several new monochromatic patterns involving additive, multiplicative, and symmetric structures. Throughout our work, we use tools from the Algebra of the Stone-\v{C}ech Compactifications of discrete semigroups.

math.CO↗

arXiv2024

Cartesian products of two $CR$ sets

The notions of CR set is intimately related with the generalized van der Waerden's theorem. In this article, we prove the product of two CR sets is again a CR set. This answers [Question 4.2., N. Hindman, H. Hosseini, D. Strauss, and M. Tootkaboni: Combinatorially rich sets in arbitrary semigroups, Semigroup Forum, 107 (2023), 127-143.] . We use combinatorial arguments to prove our result.

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