Related papers: Subsets of $\mathbb{F}^*_p$ with only small produc…

Let $F_p$ be the field of a prime order $p$. Then for any positive integer $n>1$, for any $\epsilon>0$, and for any subset $A$ of $F_p$, every element of $F_p$ can be represented as a sum of $N$ elements, each of them is a product of $n$…

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…

For any prime p we consider the density of elements in the multiplicative group of the finite field F_p having order, respectively index, congruent to a(mod d). We compute these densities on average, where the average is taken over all…

Let $p$ be a prime, let $S$ be a non-empty subset of $\mathbb{F}_p$ and let $0<\epsilon\leq 1$. We show that there exists a constant $C=C(p, \epsilon)$ such that for every positive integer $k$, whenever $\phi_1, \dots, \phi_k:…

For a given subset $A\subseteq \mathbb F_q^*$, we study the problem of finding a large packing set $B$ of $A$, that is, a set $B \subseteq \mathbb F_q^*$ such that $|AB|=|A||B|$. We prove the existence of such a $B$ of size $|B|\ge…

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…

For a polynomial $F(t,A_1,\ldots,A_n)\in\mathbf{F}_p[t,A_1,\ldots,A_n]$ ($p$ being a prime number) we study the factorization statistics of its specializations $$F(t,a_1,\ldots,a_n)\in\mathbf{F}_p[t]$$ with $(a_1,\ldots,a_n)\in S$, where…

In this paper we study the ratio between the number of $p$-elements and the order of a Sylow $p$-subgroup of a finite group $G$. As well known, this ratio is a positive integer and we conjecture that, for every group $G$, it is at least the…

We generalize the geometric sequence $\{a^p, a^{p-1}b, a^{p-2}b^2,...,b^p\}$ to allow the $p$ copies of $a$ (resp. $b$) to all be different. We call the sequence $\{a_1a_2a_3\cdots a_p, b_1a_2a_3\cdots a_p, b_1b_2a_3\cdots a_p,\ldots,…

For a large prime $p$, and a polynomial $f$ over a finite field $F_p$ of $p$ elements, we obtain a lower bound on the size of the multiplicative subgroup of $F_p^*$ containing $H\ge 1$ consecutive values $f(x)$, $x = u+1, \ldots, u+H$,…

Let $p$ be a prime and let $A$ be a subset of $\mathbb{F}_p$ with $A=-A$ and $|A\setminus\{0\}| \leq 2\log_3(p)$. Then there is an element of $\mathbb{F}_p$ which has a unique representation as a difference of two elements of $A$.

Under the generalized Riemann hypothesis, we illustrate that the ratio of the set of primes $p$ such that $\langle -1, 2 \rangle$ has an odd prime index in $\mathbb{F}_p^*$ to the set of primes $p$ such that the subgroup has index greater…

Let p be a prime and let A be a subset of F_p. For k<p let X_{A,k} be the (k-1)-dimensional complex on the vertex set F_p with a full (k-2)-skeleton whose (k-1)-faces are k-subsets S of F_p such that the sum of the elements of S belongs to…

We obtain a new lower bound on the size of value set f(F_p) of a sparse polynomial f in F_p[X] over a finite field of p elements when p is prime. This bound is uniform with respect of the degree and depends on some natural arithmetic…

We estimate the asymptotic density of the set $\bar{A}$ of primes $p$ satisfying the constraint that $p+1$ and $p-1$ have only one prime divisor larger than $3$. We also estimate the density of a maximal subset $\bar{B} \subset \bar{A}$…

There exists an absolute constant $C$ with the following property. Let $A \subseteq \mathbb{F}_p$ be a set in the prime order finite field with $p$ elements. Suppose that $|A| > C p^{5/8}$. The set \[ (A \pm A)(A \pm A) = \{(a_1 \pm…

Let $\mathbb{F}_p$ be the field of a prime order $p.$ It is known that for any integer $N\in [1,p]$ one can construct a subset $A\subset\mathbb{F}_p$ with $|A|= N$ such that $$ \max\{|A+A|, |AA|\}\ll p^{1/2}|A|^{1/2}. $$ In the present…

Let $\F_p$ be the field of residue classes modulo a large prime $p$. The present paper is devoted to the problem of representability of elements of $\F_p$ as sums of fractions of the form $x/y$ with $x,y$ from short intervals of $\F_p$.

Fix $a \in \mathbb{Z}$, $a\notin \{0,\pm 1\}$. A simple argument shows that for each $\epsilon > 0$, and almost all (asymptotically 100% of) primes $p$, the multiplicative order of $a$ modulo $p$ exceeds $p^{\frac12-\epsilon}$. It is an…

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…