Related papers: Additive decompositions of large multiplicative su…

We say that a set $S$ is additively decomposed into two sets $A$ and $B$ if $S = \{a+b : a\in A, \ b \in B\}$. A. S\'ark\"ozy has recently conjectured that the set $Q$ of quadratic residues modulo a prime $p$ does not have nontrivial…

We say that a set $S$ is additively decomposed into two sets $A$ and $B$, if $S = \{a+b : a\in A, \ b \in B\}$. Here we study additively decompositions of multiplicative subgroups of finite fields. In particular, we give some improvements…

In this paper, we employ a version of Stepanov's method, developed by Hanson and Petridis, to prove several results on additive irreducibility of multiplicative subgroups in finite fields of prime order $p$. Specifically, we show that if a…

We prove that finite sets of real numbers satisfying $|AA| \leq |A|^{1+\epsilon}$ with sufficiently small $\epsilon > 0$ cannot have small additive bases nor can they be written as a set of sums $B+C$ with $|B|, |C| \geq 2$. The result can…

The multiplicative group of a finite field is well known to be cyclic; in this note, we determine the finite fields whose multiplicative groups are direct sum indecomposable. We obtain our classification using a direct argument and also as…

In this paper, we investigate the multiplicative structure of a shifted multiplicative subgroup and its connections with additive combinatorics and the theory of Diophantine equations. Among many new results, we highlight our main…

In an earlier paper, we determined the finite fields with indecomposable multiplicative groups and conjectured that there is no infinite field whose multiplicative group is indecomposable. In this paper, we prove this conjecture for several…

Let $p\equiv1\pmod3$ be a prime . We study several topics on additive decompositions concerning the set $C_p$ of all non-zero cubes in the finite field of $p$ elements. For example, we show that when $p>184291$ , the set $C_p$ has no…

We classify those finite fields $\mathbb{F}_q$, for $q$ a power of some fixed prime number, whose members are the sum of an $n$-potent element with $n>1$ and a 4-potent element. It is shown that there are precisely ten non-trivial pairs…

We establish the restricted sumset analogue of the celebrated conjecture of S\'{a}rk\"{o}zy on additive decompositions of the set of nonzero squares over a finite field. More precisely, we show that if $q>13$ is an odd prime power, then the…

We extend a bound of Roche-Newton, Shparlinski and Winterhof which says any subset of a finite field can be decomposed into two disjoint subset $\cU$ and $\cV$ of which the additive energy of $\cU$ and $f(\cV)$ are small, for suitably…

We prove that for any finite set A of real numbers its difference set D:=A-A has large product set and quotient set, namely, |DD|, |D/D| \gg |D|^{1+c}, where c>0 is an absolute constant. A similar result takes place in the prime field F_p…

In a recent breakthrough, Kalmynin resolved conjectures of Lev--Sonn and S\'{a}rk\"{o}zy on additive decompositions of multiplicative subgroups of prime fields. In this paper, inspired by a related conjecture of S\'{a}rk\"{o}zy, we prove…

It has been conjectured by Sarkozy that with finitely many exceptions, the set of quadratic residues modulo a prime $p$ cannot be represented as a sumset $\{a+b\colon a\in A, b\in B\}$ with non-singleton sets $A,B\subset F_p$. The case…

Let $p$ and $q$ be polynomials with degree $2$ over an arbitrary field $\mathbb{F}$. In the first part of this article, we characterize the matrices that can be decomposed as $A+B$ for some pair $(A,B)$ of square matrices such that $p(A)=0$…

Let $\mathbb{F}$ be a finite field of odd characteristic. When $|\mathbb{F}|\ge 5$, we prove that every matrix $A$ admits a decomposition into $D+M$ where $D$ is diagonalizable and $M^2=0$. For $\mathbb{F}=\mathbb{F}_3$, we show that such…

In this paper, we prove a quantitative version of the statement that every nonempty finite subset of $\mathbb{N}^+$ is a set of quadratic residues for infinitely many primes of the form $[n^c]$ with $1\leqslant c\leqslant243/205$.…

Let $q$ be a power of a prime and let $\mathbb{F}_q$ be the finite field consisting of $q$ elements. We establish new explicit estimates on Gauss sums of the form $S_n(a) = \sum_{x\in \mathbb{F}_q}\psi_a(x^n)$, where $\psi_a$ is a…

Let $A$ be a simple abelian variety over a number field $k$ such that $\operatorname{End}(A)$ is noncommutative. We show that $A$ splits modulo all but finitely many primes of $k$. We prove this by considering the subalgebras of…

Let F_q be the finite field of q elements. Let H be a multiplicative subgroup of F_q^*. For a positive integer k and element b\in F_q, we give a sharp estimate for the number of k-element subsets of H which sum to b.