January 22, 2023
Similar papers 2
July 29, 2018
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 $A(A + 1)$. Our result improves on the previous best known bound due to Zhelezov and holds under more relaxed restrictions.
February 5, 2016
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 \[ (A \pm A)(A \pm A) = \{(a_1 \pm a_2)(a_3 \pm a_4) : a_1,a_2,a_3,a_4 \in A\} \] contains at least $p/2$ elements.
January 16, 2016
We compare the size of the difference set $A-A$ to that of the set $kA$ of $k$-fold sums. We show the existence of sets such that $|kA| < |A-A|^{a_k}$ with $a_k<1$.
January 1, 2006
A positive integer n is called a covering number if there are some distinct divisors n_1,...,n_k of n greater than one and some integers a_1,...,a_k such that Z is the union of the residue classes a_1(mod n_1),...,a_k(mod n_k). A covering number is said to be primitive if none of its proper divisors is a covering number. In this paper we give some sufficient conditions for n to be a (primitive) covering number; in particular, we show that for any r=2,3,... there are infinitel...
October 1, 2013
For a set $\cM=\{-\mu,-\mu+1,\ldots, \lambda\}\setminus\{0\}$ with non-negative integers $\lambda,\mu<q$ not both 0, a subset $\cS$ of the residue class ring $\Z_q$ modulo an integer $q\ge 1$ is called a $(\lambda,\mu;q)$-\emph{covering set} if $$ \cM \cS=\{ms \bmod q : m\in \cM,\ s\in \cS\}=\Z_q. $$ Small covering sets play an important role in codes correcting limited-magnitude errors. We give an explicit construction of a $(\lambda,\mu;q)$-covering set $\cS$ which is of ...
June 9, 2023
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 $xy^{-1}$ for all distinct two elements $x,y\in D$, represent each non-identity element in $G$ exactly $\lambda$ times. A $(v,k,\lambda)$-signed difference set is a generalization of a $(v,k,\lambda)$-difference set $D$, which satisfies all prope...
April 20, 2015
We provide upper bounds on the largest subsets of $\{1,2,\dots,N\}$ with no differences of the form $h_1(n_1)+\cdots+h_{\ell}(n_{\ell})$ with $n_i\in \mathbb{N}$ or $h_1(p_1)+\cdots+h_{\ell}(p_{\ell})$ with $p_i$ prime, where $h_i\in \mathbb{Z}[x]$ lie in in the classes of so-called intersective and $\mathcal{P}$-intersective polynomials, respectively. For example, we show that a subset of $\{1,2,\dots,N\}$ free of nonzero differences of the form $n^j+m^k$ for fixed $j,k\in \...
August 16, 2015
We prove, in particular, that if A,G are two arbitrary multiplicative subgroups of the prime field f_p, |G| < p^{3/4} such that the difference A-A is contained in G then |A| \ll |\G|^{1/3+o(1)}. Also, we obtain that for any eps>0 and a sufficiently large subgroup G with |G| \ll p^{1/2-eps} there is no representation G as G = A+B, where A is another subgroup, and B is an arbitrary set, |A|,|B|>1. Finally, we study the number of collinear triples containing in a set of f_p and ...
May 13, 2015
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 prove a similar result. For all suitably large primes $q$ we define $P_\eta$ to be the set of primes less than $\eta q$, viewed naturally as a subset of $\left(\mathbb{Z}/ q\mathbb{Z}\right)^{\times}$. Considering the $k$-fold product set $P_\...
December 1, 2017
We prove new results on additive properties of finite sets $A$ with small multiplicative doubling $|AA|\leq M|A|$ in the category of real/complex sets as well as multiplicative subgroups in the prime residue field. The improvements are based on new combinatorial lemmata, which may be of independent interest. Our main results are the inequality $$ |A-A|^3|AA|^5 \gtrsim |A|^{10}, $$ over the reals, "redistributing" the exponents in the textbook Elekes sum-product inequality a...