Related papers: Difference sets are not multiplicatively closed

We prove lower bounds for the number of primes $p \leq N + b$ such that $p-b$ is divisible by $2^{k(N)}$ and has at most $k$ odd prime factors ($k \geq 2$), assuming $2^{k(N)} \leq N^\theta$ for some $\theta > 0$ depending on $k$. The proof…

Let $\mathbb{F}_p$ be a prime field of order $p,$ and $A$ be a set in $\mathbb{F}_p$ with $|A| \leq p^{1/2}.$ In this note, we show that \[\max\{|A+A|, |f(A, A)|\}\gtrsim |A|^{\frac{6}{5}+\frac{4}{305}},\] where $f(x, y)$ is a…

The smallness is proved of fundamental groups for arithmetic schemes. This is a higher dimensional analogue of the Hermite-Minkowski theorem. We also refer to the case of varieties over finite fields. As an application, we prove certain…

Given a real analytic set X in a complex manifold and a positive integer d, denote by A(d) the set of points p in X at which there exists a germ of a complex analytic set of dimension d contained in X. It is proved that A(d) is a closed…

Let p be a prime number and M a quadratic number field, M not equal to Q(\sqrt{p}) if p is congruent to 1 modulo 4. We will prove that for any positive integer d there exists a Galois extension F/Q with Galois group D_{2p} and an elliptic…

Let $p$ be a prime and $G$ a subgroup of $GL_d(p)$. We define $G$ to be $p$-exceptional if it has order divisible by $p$, but all its orbits on vectors have size coprime to $p$. We obtain a classification of $p$-exceptional linear groups.…

Given a finite set $A\subset \mathbb{R}\backslash \{0\}$, define \begin{align*}&A\cdot A \ =\ \{a_i\cdot a_j\,|\, a_i,a_j\in A\},\\ &A/A \ =\ \{a_i/a_j\,|\,a_i,a_j\in A\},\\ &A + A \ =\ \{a_i + a_j\,|\, a_i,a_j\in A\},\\ &A - A \ =\ \{a_i -…

Let $\mathbb{F}_q$ be a finite field of order $q$, where $q$ is a power of a prime. For a set $A \subset \mathbb{F}_q$, under certain structural restrictions, we prove a new explicit lower bound on the size of the product set $A(A + 1)$.…

We show that, for a finite set $A$ of real numbers, the size of the set $$\frac{A+A}{A+A} = \left\{ \frac{a+b}{c+d} : a,b,c,d \in A, c+d \neq 0 \right \}$$ is bounded from below by $$\left|\frac{A+A}{A+A} \right| \gg \frac{|A|^{2+1/4}}{|A /…

We prove that there exists an absolute constant $c>0$ such that if an arithmetic progression $\cP$ modulo a prime number $p$ does not contain zero and has the cardinality less than $cp$, then it can not be represented as a product of two…

Let F be a non-Archimedean locally compact field of residue characteristic p, let G be an inner form of GL(n,F) with n>0, and let l be a prime number different from p. We describe the block decomposition of the category of finite length…

We study M-separability as well as some other combinatorial versions of separability. In particular, we show that the set-theoretic hypothesis b=d implies that the class of selectively separable spaces is not closed under finite products,…

We consider the set of monic irreducible polynomials $P$ over a finite field $\mathbb{F}_q$ such that the multiplicative order modulo $P$ of some a in $\mathbb{F}_q(T)$ is divisible by a fixed positive integer $d$. Call $R_q(a,d)$ this set.…

We prove that there is a small but fixed positive integer e such that for every prime larger than a fixed integer, every subset S of the integers modulo p which satisfies |2S|<(2+e)|S| and 2(|2S|)-2|S|+2 < p is contained in an arithmetic…

It is proved that any countable topological vector space over a finite field $\mathbb F_p$ or, equivalently, any countable Abelian topological group of prime exponent has a closed discrete basis.

Let $\varepsilon>0$ be a fixed small constant, ${\mathbb F}_p$ be the finite field of $p$ elements for prime $p$. We consider additive and multiplicative problems in ${\mathbb F}_p$ that involve intervals and arbitrary sets. Representative…

Let $X$ be a scheme of finite type over $\mathbf{Z}$. For $p \in \mathcal{P}$ the set of prime numbers, let $N_{X}(p)$ be the number of $\mathbf{F}_{p}$-points of $X/\mathbf{F}_{p}$. For fixed $n\geq 1$ and $a_{1}, \ldots, a_{n} \in…

Generalising a previous result, we determine all non-abelian finite simple groups whose order has largest prime divisor not exceeding $10^4$. The computer code for this and similar calculations is made available.

Signed difference sets have interesting applications in communications and coding theory. A $(v,k,\lambda)$-difference set in a finite group $G$ of order $v$ is a subset $D$ of $G$ with $k$ distinct elements such that the expressions…

It is proved that, in certain subgroups of direct products of countable groups, the property of being an unconditionally closed set coincides with that of being an algebraic set. In particular, these properties coincide in all Abelian…