Related papers: The Freiman--Ruzsa Theorem over Finite Fields

Let A be a commutative noetherian ring. Call a functor <<commutative A-algebras>> --> <<sets>> coherent if it can be built up (via iterated finite limits) from functors of the form B \mapsto M tensor_A B, where M is a f.g. A-module. When…

In this paper we prove that every set $A\subset\mathbb{Z}$ satisfying the inequality $\sum_{x}\min(1_A*1_A(x),t)\le(2+\delta)t|A|$ for $t$ and $\delta$ in suitable ranges, then $A$ must be very close to an arithmetic progression. We use…

We consider gauge theories defined in higher dimensions where the extra dimensions form a fuzzy space (a finite matrix manifold). We reinterpret these gauge theories as four-dimensional theories with Kaluza-Klein modes. We then perform a…

By a [$K$-]approximate subring of a ring we mean an additively symmetric subset $X$ such that $X \cdot X \cup (X + X)$ is covered by finitely many [resp.\ $K$] additive translates of $X$. We prove a structure theorem for finite approximate…

Erd\H{o}s, Gallai, and Tuza posed the following problem: given an $n$-vertex graph $G$, let $\tau_1(G)$ denote the smallest size of a set of edges whose deletion makes $G$ triangle-free, and let $\alpha_1(G)$ denote the largest size of a…

Let G be a finite group with exactly k elements of largest possible order m. Let q(m) be the product of gcd(m,4) and the odd prime divisors of m. We show that |G|\le q(m)k^2/\phi(m) where \phi denotes Euler's totient function. This…

For a finite group $A$ with normal subgroup $G$, a subgroup $U$ of $G$ is an $A$-prime-power-covering subgroup if $U$ meets every $A$-conjugacy-class of elements of $G$ of prime power order. It is conjectured that $|G:U|$ is bounded by some…

Let $A$ be the product of an abelian variety and a torus over a number field $K$, and let $m$ be a positive integer. If $\alpha \in A(K)$ is a point of infinite order, we consider the set of primes $\mathfrak p$ of $K$ such that the…

Let $T$ be a (first order complete) dependent theory, ${\mathfrak{C}}$ a $\bar\kappa$-saturated model of $T$ and $G$ a definable subgroup which is abelian. Among subgroups of bounded index which are the union of $<\bar\kappa$ type definable…

An affine variety $X$ of dimension $\ge 2$ is called {\em flexible} if its special automorphism group SAut$(X)$ acts transitively on the smooth locus $X_{reg}$ \cite{AKZ}. Recall that the special automorphism group SAut$(X)$ is the subgroup…

We study sum-free sets in sparse random subsets of even order abelian groups. In particular, we determine the sharp threshold for the following property: the largest such set is contained in some maximum-size sum-free subset of the group.…

Let $F(G)$ and $b(G)$ respectively denote the Fitting subgroup and the largest degree of an irreducible complex character of a finite group $G$. A well-known conjecture of D. Gluck claims that if $G$ is solvable then $|G:F(G)|\leq…

Let $K_{q^n}(a)$ be a Kloosterman sum over the finite field $\F_{q^n}$ of characteristic $p$. In this note so called subfield conjecture is proved in case $p>3$: if $a\ne0$ belongs to the proper subfield $\F_q$ of $\F_{q^n}$, then…

Let $k$ be a perfect field such that for every $n$ there are only finitely many field extensions, up to isomorphism, of $k$ of degree $n$. If $G$ is a reductive algebraic group defined over $k$, whose characteristic is very good for $G$,…

Let A, B and S be three subsets of a finite Abelian group G. The restricted sumset of A and B with respect to S is defined as A\wedge^{S} B= {a+b: a in A, b in B and a-b not in S}. Let L_S=max_{z in G}| {(x,y): x,y in G, x+y=z and x-y in…

The Grassmann convexity conjecture gives a conjectural formula for the maximal total number of real zeros of the consecutive Wronskians of an arbitrary fundamental solution to a disconjugate linear ordinary differential equation with real…

We note a parallel between some ideas of stable model theory and certain topics in finite combinatorics related to the sum-product phenomenon. For a simple linear group G, we show that a finite subset X with |X X \^{-1} X |/ |X| bounded is…

Let $\delta > 1/2$. We prove that if $A$ is a subset of the primes such that the relative density of $A$ in every reduced residue class is at least $\delta$, then almost all even integers can be written as the sum of two primes in $A$. The…

Consider any Dirichlet series sum a_n/n^z with nonnegative coefficients a_n and finite sum function f(z)=f(x+iy) when x is greater than 1. Denoting the partial sum a_1+...+a_N by s_N, the paper gives the following necessary and sufficient…

Given a number field $F$ with ring of integers $\mathcal{O}_{F}$, one can associate to any torsion free subgroup of $\operatorname{SL}(2,\mathcal{O}_{F})$ of finite index a complete Riemannian manifold of finite volume with fibered cusp…