Related papers: Popular Differences for Corners in Abelian Groups

The Corners Theorem states that for any $\alpha > 0$ there exists an $N_0$ such that for any abelian group $G$ with $|G| = N \geq N_0$ and any subset $A \subset G \times G$ with $|A| \ge \alpha N^2$ we can find a corner in $A$ , i.e. there…

Two versions of the Ajtai-Szemer\'edi Theorem are considered in the Cartesian square of a finite non-Abelian group $G$. In case $G$ is sufficiently quasirandom, we obtain strong forms of both versions: if $E \subseteq G\times G$ is fairly…

Let G be a finite Abelian group and A be a subset G\times G of cardinality at least |G|^2/(log log |G|)^c, where c>0 is an absolute constant. We prove that A contains a triple {(k,m), (k+d,m), (k,m+d)}, where d does not equal 0. This…

We prove a conjecture due to Baumgaertel and Lledo according to which for every compact group G one has Z(G)^ \cong C(G), where the `chain group' C(G) is the free abelian group (written multiplicatively) generated by the set G^ of…

Let $G$ be a finite abelian group and $A$ be a subset of $G \times G$ which is corner--free, meaning that there are no $x, y \in G$ and $d \in G \setminus \{0\}$ such that $(x, y)$, $(x+d, y)$, $(x, y+d) \in A$. We prove that \[|A| \le…

Let $G$ be a finite $D$-quasirandom group and $A \subset G^{k}$ a $\delta$-dense subset. Then the density of the set of side lengths $g$ of corners \[ \{(a_{1},\dots,a_{k}),(ga_{1},a_{2},\dots,a_{k}),\dots,(ga_{1},\dots,ga_{k})\} \subset A…

Let $\A$ be a ring with local units, $\E$ a set of local units for $\A$, $\G$ an abelian group and $\alpha$ a partial action of $\G$ by ideals of $\A$ that contain local units and such that the partial skew group ring $\A\star_{\alpha} \G$…

Given a finite abelian group $G$ and cyclic subgroups $A$, $B$, $C$ of $G$ of the same order, we find necessary and sufficient conditions for $A$, $B$, $C$ to admit a common transversal for the cosets they afford. For an arbitrary number of…

Let $G$ be a finite group and $\alpha(G)=\frac{|C(G)|}{|G|}$\,, where $C(G)$ denotes the set of cyclic subgroups of $G$. In this short note, we prove that $\alpha(G)\leq\alpha(Z(G))$ and we describe the groups $G$ for which the equality…

In an additive group (G,+), a three-dimensional corner is the four points g, g+d(1,0,0), g+d(0,1,0), g+d(0,0,1), where g is in G^3, and d is a non-zero element of G. The Ramsey number of interest is R_3(G) the maximal cardinality of a…

For an abelian topological group G let G^* denote the dual group of all continuous characters endowed with the compact open topology. Given a closed subset X of an infinite compact abelian group G such that w(X) < w(G) and an open…

Szemeredi's regularity lemma is an important tool in graph theory which has applications throughout combinatorics. In this paper we prove an analogue of Szemeredi's regularity lemma in the context of abelian groups and use it to derive some…

Let G be any abelian group and {a_sG_s}_{s=1}^k be a finite system of cosets of subgroups G_1,...,G_k. We show that if {a_sG_s}_{s=1}^k covers all the elements of G at least m times with the coset a_tG_t irredundant then [G:G_t]\le 2^{k-m}…

We show that a compact group $G$ has finite conjugacy classes, i.e., is an FC-group if and only if its center $Z(G)$ is open if and only if its commutator subgroup $G'$ is finite. Let $d(G)$ denote the Haar measure of the set of all pairs…

Given a group G, a (unital) ring A and a group homomorphism $\sigma : G \to \Aut(A)$, one can construct the skew group ring $A \rtimes_{\sigma} G$. We show that a skew group ring $A \rtimes_{\sigma} G$, of an abelian group G, is simple if…

It has been recently proved (by Croot, Lev and Pach and the subsequent work by Ellenberg and Gijswijt) that for a group $G=G_0^n$, where $G_0\ne \{1,-1\}^m$ is a fixed finite Abelian group and $n$ is large, any subset $A$ without…

We extend the class of abelian groups for which a conjecture of Asai and Yoshida on the number of crossed homomorphisms holds. We also prove a general result which connects certain problems concerning divisibility in groups to the…

Suppose that $S \subseteq [n]^2$ contains no three points of the form $(x,y), (x,y+\delta), (x+\delta,y')$, where $\delta \neq 0$. How big can $S$ be? Trivially, $n \le |S| \le n^2$. Slight improvements on these bounds are obtained from…

The power graph of a group $G$, denoted as $P(G)$, constitutes a simple undirected graph characterized by its vertex set $G$. Specifically, vertices $a,b$ exhibit adjacency exclusively if $a$ belongs to the cyclic subgroup generated by $b$…

A group $G$ is said to be $\frac{3}{2}$-generated if every nontrivial element belongs to a generating pair. It is easy to see that if $G$ has this property then every proper quotient of $G$ is cyclic. In this paper we prove that the…