相关论文: Growth of small generating sets in SL_n(Z/pZ)

Let G=SL_3(Z/pZ), p a prime. Let A be a set of generators of G. Then A grows under the group operation. To be precise: denote by |S| the number of elements of a finite set S. Assume |A| < |G|^{1-\epsilon} for some \epsilon>0. Then |A\cdot…

Let $K=Z/pZ$ and let $A$ be a subset of $\GL_r(K)$ such that $<A>$ is solvable. We reduce the study of the growth of $A$ under the group operation to the nilpotent setting. Specifically we prove that either $A$ grows rapidly (meaning…

We show that every subset of SL_2(Z/pZ) grows rapidly when it acts on itself by the group operation. It follows readily that, for every set of generators A of SL_2(Z/pZ), every element of SL_2(Z/pZ) can be expressed as a product of at most…

In a previous paper of the authors, we showed that for any polynomials $P_1,\dots,P_k \in \Z[\mathbf{m}]$ with $P_1(0)=\dots=P_k(0)$ and any subset $A$ of the primes in $[N] = \{1,\dots,N\}$ of relative density at least $\delta>0$, one can…

We prove that if L is a finite simple group of Lie type and A a symmetric set of generators of L, then A grows i.e |AAA| > |A|^{1+epsilon} where epsilon depends only on the Lie rank of L, or AAA=L. This implies that for a family of simple…

We consider the problem of sums of dilates in groups of prime order. We show that given $A\subset \Z{p}$ of sufficiently small density then $$\big| \lambda_{1}A+\lambda_{2}A+...+ \lambda_{k}A \big|…

In this paper we show that if $A$ is a subset of the primes with positive relative density $\delta$, then $A+A$ must have positive upper density $C_1\delta e^{-C_2(\log(1/\delta))^{2/3}(\log\log(1/\delta))^{1/3}}$ in $\mathbb{N}$. Our…

Let $\beta,\epsilon \in (0,1]$, and $k \geq \exp(122 \max\{1/\beta,1/\epsilon\})$. We prove that if $A,B$ are subsets of a prime field $\mathbb{Z}_{p}$, and $|B| \geq p^{\beta}$, then there exists a sum of the form $$S = a_{1}B \pm \ldots…

Answering a question of T. Cochrane and C. Pinner, we prove that for any {\epsilon}>0, sufficiently large prime number p and an arbitrary multiplicative subgroup R of the field Z/pZ, p^{\epsilon} < |R| < p^{2/3-{\epsilon}} the following…

Let $G$ be a $p$-group for some prime $p$. Let $n$ be the positive integer so that $|G:Z(G)| = p^n$. Suppose $A$ is a maximal abelian subgroup of $G$. Let $$p^l = {\rm max} \{|Z(C_G (g)):Z(G)| : g \in G \setminus Z(G)\},$$ $$p^b = {\rm max}…

The classical theorem of Schnirelmann states that the primes are an additive basis for the integers. In this paper we consider the analogous multiplicative setting of the cyclic group $\left(\mathbb{Z}/ q\mathbb{Z}\right)^{\times}$, and…

Let $R$ be a ring generated by $l$ elements with stable range $r$. Assume that the group $EL_d(R)$ has Kazhdan constant $\epsilon_0>0$ for some $d > r $. We prove that there exist $\epsilon(\epsilon_0,l) >0$ and $k \in N$, s.t. for every $n…

We show that if A is a subset of {1, ..., n} such that it has no pairs of elements whose difference is equal to p-1 with p a prime number, then the size of A is O(n(loglog n)^(-clogloglogloglog n)) for some positive constant c.

We prove that if L is a finite simple group of Lie type and A a symmetric set of generators of L, then A grows i.e |AAA| > |A|^(1+epsilon) where epsilon depends only on the Lie rank of L, or AAA=L. This implies that for a family of simple…

For the sequence defined by a(n) = a(n-1) + gcd(n, a(n-1)) with a(1) = 7 we prove that a(n) - a(n-1) takes on only 1s and primes, making this recurrence a rare "naturally occurring" generator of primes. Toward a generalization of this…

Let $f(n)=\min_{p} |n-p|$, where $p$ is a prime. We show that there is a positive constant $\delta$ such that for any large integer $N$ there exist two positive integers $n_1$ and $n_2$ such that $N=n_1 + n_2$ and $f(n_i)\gg \ln N (\ln\ln…

We prove that every set $A\subset\mathbb{Z}/p\mathbb{Z}$ with $\mathbb{E}_x\min(1_A*1_A(x),t)\le(2+\delta)t\mathbb{E}_x 1_A(a)$ is very close to an arithmetic progression. Here $p$ stands for a large prime and $\delta,t$ are small real…

Let $\{p_j(n)\}_{j=1}^{\omega(n)}$ denote the increasing sequence of distinct prime factors of an integer $n$. For $z\geqslant 0$, let $G(n;z)$ denote the number of those indexes $j$ such that $p_{j+1}(n)>p_j(n)^{\exp z}$. We show uniform…

When $k>1$ and $n$ is the product of the smallest $k$ primes, the $(k+1)$-st smallest prime is the least divisor exceeding $1$ of $n^{n^n}-1$. This variant of Euclid's prime generator is discussed with some of its cousins.

For $p$ being a large prime number, and $A \subset \mathbb{F}_p$ we prove the following: $(i)$ If $A(A+A)$ does not cover all nonzero residues in $\mathbb{F}_p$, then $|A| < p/8 + o(p)$. $(ii)$ If $A$ is both sum-free and satisfies $A =…