Related papers: Veech's theorem of higher order

We present a purely enveloping semigroup proof of a theorem of Shao and Ye which asserts that for an abelian group $T$, a minimal flow $(X,T)$ and any integer $d \ge 1$, the regional proximal relation of order $d$ is an equivalence…

We introduce higher order regionally proximal relations suitable for an arbitrary acting group. For minimal abelian group actions, these relations coincide with the ones introduced by Host, Kra and Maass. Our main result is that these…

For $d \geq 2$ and $G$ a finite abelian group, define $T_d(G)$ to be the minimum number of vertices $n$ so that there exists a simplicial complex $X$ on $n$ vertices which has the torsion part of $H_{d - 1}(X)$ isomorphic to $G$. Here we…

Let G be a connected reductive group defined over an algebraically closed field k of characteristic p > 0. The purpose of this paper is two-fold. First, when p is a good prime, we give a new proof of the ``order formula'' of D. Testerman…

For a finite abelian group $G$, let $\beta_{\mathrm{sep}}(G)$ denote its separating Noether number. We determine $\beta_{\mathrm{sep}}(G)$ exactly for every finite abelian group $ G \cong C_{n_1}\oplus \cdots \oplus C_{n_r}$ with $ 1<n_1…

A result of Pyber states that every finite group $G$ contains an abelian subgroup whose order is quasi-polynomially large in $\lvert G\rvert$. We prove a similar result for $K$-approximate subgroups of solvable groups under only modest…

Let $G$ be a periodic group, and let $LCM(G)$ be the set of all $x\in G$ such that $o(x^nz)$ divides the least common multiple of $o(x^n)$ and $o(z)$ for all $z$ in $G$ and all integers $n$. In this paper, we prove that the subgroup…

Given a finite group $G$, we denote by $\psi\,'(G)$ the product of element orders of $G$. Our main result proves that the restriction of $\psi\,'$ to abelian $p$-groups of order $p^n$ is strictly increasing with respect to a natural order…

By proving the minimality of face transformations acting on the diagonal points and searching the points allowed in the minimal sets, it is shown that the regionally proximal relation of order $d$, $\RP^{[d]}$, is an equivalence relation…

For a finite abelian group $G$ and a positive integer $d$, let $\mathsf s_{d \mathbb N} (G)$ denote the smallest integer $\ell \in \mathbb N_0$ such that every sequence $S$ over $G$ of length $|S| \ge \ell$ has a nonempty zero-sum…

For a finite abelian group $G$ and a positive integer $h$, the unrestricted (resp.~restricted) $h$-critical number $\chi(G,h)$ (resp.~$\chi \hat{\;}(G,h)$) of $G$ is defined to be the minimum value of $m$, if exists, for which the $h$-fold…

Erd\H os and R\'{e}nyi claimed and Vu proved that for all $h \ge 2$ and for all $\epsilon > 0$, there exists $g = g_h(\epsilon)$ and a sequence of integers $A$ such that the number of ordered representations of any number as a sum of $h$…

Let $\ell$ and $m$ be positive integers with $\ell \leq m$, and let $\mathcal{A} = (A_1, \ldots, A_m)$ be a finite sequence of finite subsets of a group $G$ (not necessarily abelian), written multiplicatively. The {\it generalized product…

Given a group $G$ with bounded torsion that acts properly on a systolic complex, we show that every solvable subgroup of $G$ is finitely generated and virtually abelian of rank at most $2$. In particular this gives a new proof of the above…

We study several aspects of higher-order regionally proximal relations for group actions. First, we develop an algebraic approach to study higher-order regionally proximal relations. To this end, we introduce a new topology on a subgroup of…

Veech's Theorem claims that if $G$ is a locally compact\,(LC) Hausdorff topological group, then it may act freely on $G^{\textrm{LUC}}$. We prove Veech's Theorem for $G$ being only locally quasi-totally bounded, not necessarily LC. And we…

Let $k$ be a field, $G$ be an abelian group and $r\in \mathbb N$. Let $L$ be an infinite dimensional $k$-vector space. For any $m\in End_k(L)$ we denote by $r(m)\in [0,\infty ]$ the rank of $m$. We define by $R(G,r,k)\in [0,\infty]$ the…

Let $G$ be a multiplicatively written finite group. We denote by $\mathsf E(G)$ the smallest integer $t$ such that every sequence of $t$ elements in $G$ contains a product-one subsequence of length $|G|$. In 1961, Erd\H{o}s, Ginzburg and…

Let $G$ be a dp-minimal group; we prove some consequences of several different hypotheses on $G$. First, if $G$ is torsion-free, then it is abelian. Second, if $G$ admits a distal f-generic type, then it is virtually nilpotent; we prove…

In this paper we prove that if $X $ is a Banach space, then for every lower semi-continuous bounded below function $f, $ there exists a $\left(\varphi_1, \varphi_2\right)-$convex function $g, $ with arbitrarily small norm, such that $f + g…