Related papers: Perfect difference sets constructed from Sidon set…

Let $n \in \mathbb{Z}_{\geqslant 2}$. By $P(n)$ we denote the set of all prime divisors of the integers in the sequence $n, n^2-1, (n^2-1)^2-1, \dots$. We ask whether the set $P(n)$ determines $n$ uniquely under the assumption that $n \neq…

A finite set $S \subset \mathbb{Z}$ is a Sidon set if its pairwise differences are distinct. Recall that a perfect difference set (PDS) of order $n$ is a set $B \subset \mathbb{Z}_v$ ($v = n^2 - n + 1$) of size $n$ such that every nonzero…

We prove a quantitative version of the Polynomial Szemeredi Theorem for difference sets. This result is achieved by first establishing a higher dimensional analogue of a theorem of Sarkozy (the simplest non-trivial case of the Polynomial…

We define a sequence of positive integers recursively, where each term is determined as follows: starting with a given positive integer, if the term is odd, the next is the sum of its positive divisors; if the term is even, the subsequent…

Let $v$ be a positive odd integer. A $(v,k,\lambda)$-perfect difference family (PDF) is a collection $\mathcal{F}$ of $k$-subsets of $\{0,1,\ldots,v-1\}$ such that the multiset $\bigcup_{F\in \mathcal{F}}\{x-y : x,y\in F, x>y\}$ covers each…

Two sets $A,B$ of positive integers are called \emph{exact additive complements}, if $A+B$ contains all sufficiently large integers and $A(x)B(x)/x\rightarrow1$. Let $A=\{a_1<a_2<\cdots\}$ be a set of positive integers. Denote $A(x)$ by the…

A set $A$ is dually Dedekind finite if every surjection from $A$ onto $A$ is injective; otherwise, $A$ is dually Dedekind infinite. An amorphous set is an infinite set that cannot be partitioned into two infinite subsets. A strictly…

For a nonzero integer n, a set of m distinct nonzero integers {a_1,a_2,...,a_m} such that a_i a_j + n is a perfect square for all 1 <= i < j <= m, is called a D(n)-m-tuple. In this paper, by using properties of so-called regular Diophantine…

Difference sets are subsets of a group satisfying certain combinatorial property with respect to the group operation. They can be characterized using an equality in the group ring of the corresponding group. In this paper, we exploit the…

We study the set $\mathcal{S}$ of odd positive integers $n$ with the property ${2n}/{\sigma(n)} - 1 = 1/x$, for positive integer $x$, i.e., the set that relates to odd perfect and odd "spoof perfect" numbers. As a consequence, we find that…

In this work we show that the prime distribution is deterministic. Indeed the set of prime numbers P can be expressed in terms of two subsets of N using three specific selection rules, acting on two sets of prime candidates. The prime…

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 analogy with the 290-Theorem of Bhargava-Hanke, a criterion set is a finite subset $C$ of the totally positive integers in a given totally real number field such that if a quadratic form represents all elements of $C$, then it…

We show that if $A=\{a_1 < a_2 < \ldots < a_k\}$ is a set of real numbers such that the differences of the consecutive elements are distinct, then for and finite $B \subset \mathbb{R}$, $$|A+B|\gg |A|^{1/2}|B|.$$ The bound is tight up to…

A totally symmetric set is a subset of a group such that every permutation of the subset can be realized by conjugation in the group. The (non-)existence of large totally symmetric sets obstruct homomorphisms, so bounds on the sizes of…

A symmetric tensor, which has a symmetric nonnegative decomposition, is called a completely positive tensor. We consider the completely positive tensor decomposition problem. A semidefinite algorithm is presented for checking whether a…

We prove theorems of the following form: if $A\subseteq {\mathbb R}^2$ is a big set, then there exists a big set $P\subseteq {\mathbb R}$ and a perfect set $Q\subseteq {\mathbb R}$ such that $P\times Q\subseteq A$. We discuss cases where…

Let $A$ be an infinite set of nonnegative integers. For $h \geq 2$, let $hA$ be the set of all sums of $h$ not necessarily distinct elements of $A$. If every sufficiently large integer in the sumset $hA$ has at least two representations,…

The ratio set of a set of positive integers $A$ is defined as $R(A) := \{a / b : a, b \in A\}$. The study of the denseness of $R(A)$ in the set of positive real numbers is a classical topic and, more recently, the denseness in the set of…

A discrete set in the Euclidian space is almost periodic, if the measure with the unite masses at points of the set is almost periodic in the weak sense. We prove the following result: if A is a discrete almost periodic set and the set A-A…