Related papers: Addition theorems in partially ordered groups
We study group extensions of Finite Abelian Groups using matrices. We also prove a Theorem for equivalence of extensions using matrices.
We extend Borel's theorem on the dominance of word maps from semisimple algebraic groups to some perfect groups. In another direction, we generalize Borel's theorem to some words with constants. We also consider the surjectivity problem for…
We develop an elementary theory of partially additive rings as a foundation of ${\mathbb F}_1$-geometry. Our approach is so concrete that an analog of classical algebraic geometry is established very straightforwardly. As applications, (1)…
We prove a version of the fundamental theorems of Morse Theory in the setting of finite spaces or partially ordered sets. By using these results we extend Forman's discrete Morse theory to more general cell complexes and derive the…
We combine the fixed-order evaluation of the $b\bar{b}$ sum rules with a non-relativistic effective-theory approach. The combined result for the $n$-th moment includes all terms suppressed with respect to the leading-order result by ${\cal…
For a finite abelian group $G$ with subsets $A$ and $B$, the sumset $AB$ is $\{ab \mid a\in A, b \in B\}$. A fundamental problem in additive combinatorics is to find a lower bound for the cardinality of $AB$ in terms of the cardinalities of…
Existence of long arithmetic progression in sumsets and subset sums has been studied extensively in the field of additive combinatorics. These additive combinatorics results play a central role in the recent progress of fundamental problems…
The notion of essential submodules and essential extensions of modules are extended to groups (typically nonabelian), and several necessary and sufficient conditions for a group to possess a proper essential subgroup are investigated.…
For a partial order $\preceq$ on a set X and an equivalency relation S defined on the same set X we derive a necessary and sufficient condition for the existence of such a total preorder on X whose asymmetric part contains the asymmetric…
Condensed mathematics, developed by Clausen and Scholze over the last few years, is a new way of studying the interplay between algebra and geometry. It replaces the concept of a topological space by a more sophisticated but better-behaved…
This paper provides a canonical construction of a Noetherian least fixed point topology. While such least fixed point are not Noetherian in general, we prove that under a mild assumption, one can use a topological minimal bad sequence…
The paper investigates possible generalisations of Maharam's theorem to a classification of Boolean algebras that support a finitely additive measure. We prove that Boolean algebras that support a finitely additive non-atomic uniformly…
Suppose that G is a finite group and A is a subset of G such that 1_A has algebra norm at most M. Then 1_A is a plus/minus sum of at most L cosets of subgroups of G, and L can be taken to be triply tower in O(M). This is a quantitative…
Recently, Grynkiewicz et al. [{\it Israel J. Math.} {\bf 193} (2013), 359--398], using tools from additive combinatorics and group theory, proved necessary and sufficient conditions under which the linear congruence $a_1x_1+\cdots…
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…
We prove that if $G$ is an Abelian group and $A_1,\ldots,A_k \subseteq G$ satisfy $m A_i=G$ (the $m$-fold sumset), then $A_1+\ldots+A_k=G$ provided that $k \ge c_m \log n$. This generalizes a result of Alon, Linial, and Meshulam [Additive…
Our result contains as special cases the Frobenius theorem (1895) on the~number of solutions to the equation $x^n=1$ in a finite group and the Solomon theorem (1969) on the number of solutions in a group to systems of equations with fewer…
As mathematical induction is applied to prove statements on natural numbers, {\it continuous induction} (or, {\it real induction}) is a tool to prove some statements in real analysis.(Although, this comparison is somehow an overstatement.)…
We give a short proof of Chevalley's theorem that every algebraic group is an extension of an Abelian variety by a linear algebraic group. Along the way we treat Bertini's irreducibility theorem.
The additive energy plays a central role in combinatorial number theory. We show an uncertainty inequality which indicates how the additive energy of support of a Boolean function, its degree and subcube partition are related.