Related papers: The Interplay between Additive and Multiplicative …
In analogy to the topological entropy for continuous endomorphisms of totally disconnected locally compact groups, we introduce a notion of topological entropy for continuous endomorphisms of locally linearly compact vector spaces. We study…
We prove structural results for measure preserving systems, called Furstenberg systems, naturally associated with bounded multiplicative functions. We show that for all pretentious multiplicative functions these systems always have rational…
We introduce the notion of a rational dynamical system extending the classical notion of a topological dynamical system and we prove (multiple) recurrence results for such systems via a partition theorem for the rational numbers proved by…
We make progress on two interrelated problems at the intersection of geometric measure theory, additive combinatorics and harmonic analysis: the discretised sum-product problem, and the dimension of Furstenberg sets. Along the way, we…
Central configurations have been of great interest over many years, with the earliest examples due to Euler and Lagrange. There are numerous results in the literature demonstrating the existence of central configurations with specific…
A structural analysis of construction schemes is developed. That analysis is used to give simple and new constructions of combinatorial objects which have been of interest to set theorists and topologists. We then continue the study of…
The notion of central stability was first formulated for sequences of representations of the symmetric groups by Putman. A categorical reformulation was subsequently given by Church, Ellenberg, Farb, and Nagpal using the notion of…
This paper establishes a combinatorial central limit theorem for stratified randomization, which holds under a Lindeberg-type condition. The theorem allows for an arbitrary number or sizes of strata, with the sole requirement being that…
We introduce cs-topologies, or topologies of open complemented subsets, as a new approach to constructive topology that preserves the duality between open and closed subsets of classical topology. Complemented subsets were used successfully…
In this article, we investigate the image and preimage of the important combinatorial sets such as central sets, $C$-sets, and $J_\delta$-sets which play an important role in the study of combinatorics under certain partial semigroup…
One of the many theorems Freiman proved, in the second half of the twentieth century, in the subject which later came to be known as "structure theory of set addition", was 'Freiman's $3k-4$ theorem' for subsets of $\Z$. In this article we…
This is an introductory survey on cluster algebras and their (additive) categorification using derived categories of Ginzburg algebras. After a gentle introduction to cluster combinatorics, we review important examples of coordinate rings…
Combinatorial methods (or methods of elementary transformations) came to group theory from low-dimensional topology in the beginning of the century. Soon after that, combinatorial group theory became an independent area with its own…
We give a lower bound for the size of a subset of $\mathbb F_q^n$ containing a rich k-plane in every direction, a k-plane Furstenberg set. The chief novelty of our method is that we use arguments on non-reduced subschemes and flat families…
We prove a structural result for measure preserving systems naturally associated with any finite collection of multiplicative functions that take values on the complex unit disc. We show that these systems have no irrational spectrum and…
Many clustering schemes are defined by optimizing an objective function defined on the partitions of the underlying set of a finite metric space. In this paper, we construct a framework for studying what happens when we instead impose…
Heaps of pieces were introduced by Viennot and have applications to algebraic combinatorics, theoretical computer science and statistical physics. In this paper, we how certain combinatorial properties of heaps studied by Fan and by…
Sets satisfying Central sets theorem and other Ramsey theoretic large sets were studied extensively in literature. Hindman and Strauss proved that product of some of these large sets is again large. In this paper we show that if we take two…
In 2015, Phulara established a generalization of the famous central set theorem by an original idea. Roughly speaking, this idea extends a combinatorial result from one large subset of the given semigroup to countably many. In this paper,…
We introduce a method for constructing collections of subsets of $\mathbb{R}^{n}$, using an iterated function system, a set $T,$ and a cost function. We refer to these collections as tilings. The special case where $T$ is the central open…