Research
Research
Research
Research
509+ open-access research outputs.
Let $B$ be the set of odd integers that are sums of two coprime squares. We prove that the trigonometric polynomial $S(\alpha;N)=\sum_{b\in B,b\leq N} e(b\alpha)$ satisfies \[ \frac{S(\alpha; N)}{N/\s…
The principle of maximum entropy (MaxEnt) applies to the canonical ensemble related to the number of particles, known as the $\mathcal{N}$-ensemble. This concept pertains to physical domains (or basin…
In this survey article, we explore a central theme in Diophantine approximation inspired by a celebrated result of Besicovitch on the Hausdorff dimension of well approximable real numbers. We outline …
We present constructions regarding the general behaviour of biased positional games, and amongst others show that the outcome of such a game can differ in an arbitrary way depending on which player st…
We show that once $\theta>17/30$, every sufficiently long interval $[x,x+x^\theta]$ contains many $k$-term arithmetic progressions of primes, uniformly in the starting point $x$. More precisely, for e…
The mass transference principle of Beresnevich and Velani is a powerful mechanism for determining the Hausdorff dimension/measure of $\limsup$ sets that arise naturally in Diophantine approximation. H…
In [Compositio Math. 155 (2019)] Kleinbock and Wadleigh proved a "zero-one law" for uniform inhomogeneous Diophantine approximations. We generalize this statement with arbitrary weight functions and e…
We consider approximation properties of real points by uniformly distributed sequences. Under some assumptions on the approximation functions, we prove a Khintchine-type $0$-$1$ dichotomy law. We esta…
We develop a new transference method for completely bounded $L_p$-Fourier multipliers via proper cocycles arising from probability measure-preserving group actions. This method extends earlier results…
We prove new cases of reasonable bounds for the polynomial Szemer\'{e}di theorem both over $\mathbb{Z}/N\mathbb{Z}$ with $N$ prime and over the integers. In particular, we prove reasonable bounds for …
We study the transference through finite index extensions of the notion of equational coherence, as well as its effective counterpart. We deduce an explicit algorithm for solving the following algorit…
By Maynard's theorem and the subsequent improvements by the Polymath Project, there exists a positive integer $b\leq 246$ such that there are infinitely many primes $p$ such that $p+b$ is also prime. …
Learning-based approaches, particularly reinforcement learning (RL), have become widely used for developing control policies for autonomous agents, such as locomotion policies for legged robots. RL tr…
The classical Khintchine--Jarn\'ik Theorem provides elegant criteria for determining the Lebesgue measure and Hausdorff measure of sets of points approximated by rational points, which has inspired mu…
In this paper we develop a metric theory of inhomogeneous Diophantine approximation for the case of a fixed matrix. We use transference principle to connect uniform Diophantine properties of a pair $(…
Multi-modal time series analysis has recently emerged as a prominent research area in data mining, driven by the increasing availability of diverse data modalities, such as text, images, and structure…
In this digital age, ensuring the security of digital data, especially the image data is critically important. Image encryption plays an important role in securing the online transmission/storage of i…
Confirming a conjecture of Erd\H{o}s on the chromatic number of Kneser hypergraphs, Alon, Frankl and Lov\'asz proved that in any $q$-colouring of the edges of the complete $r$-uniform hypergraph, ther…
Let ${\mathcal M}$ be a von Neumann algebra equipped with a normal semifinite faithful (nsf) trace. We say that an operator $T :{\mathcal M}\to {\mathcal M}$ is absolutely dilatable if there exist ano…
We build a bridge from density combinatorics to dimension theory of continued fractions. We establish a fractal transference principle that transfers common properties of subsets of $\mathbb N$ with p…
Free open-access publishing with Google Scholar indexing.
Submission Guide →