Related papers: On various restricted sumsets

A variety of universal algebras is called limit if it is non-finitely based but all its proper subvarieties are finitely based. Until recently, only two explicit examples of limit varieties of monoids, constructed by Jackson, were known.…

We consider a subset of projective space over a finite field and give bounds on the minimal degree of a non-vanishing form with respect to this subset.

Consider the linear congruence equation $${a_1^{s}x_1+\ldots+a_k^{s} x_k \equiv b\,(\text{mod } n^s)}\text { where } a_i,b\in\mathbb{Z},s\in\mathbb{N}$$ Denote by $(a,b)_s$ the largest $l^s\in\mathbb{N}$ which divides $a$ and $b$…

We show that for any coprime integers $\lambda_1 , \ldots , \lambda_k$ and any finite $A \subset \mathbb{Z}$, one has $$|\lambda_1 \cdot A + \ldots + \lambda_k \cdot A| \geq (|\lambda_1| + \ldots + |\lambda_k|)|A|- C,$$ where $C$ only…

We describe all sets $A \subseteq \F_p$ which represent the quadratic residues $R \subseteq \F_p$ as $R=A+A$ and $R=A\hat{+} A$. Also, we consider the case of an approximate equality $R \approx A+A$ and $R \approx A\hat{+} A$ and prove that…

In this note we prove results of the following types. Let be given distinct complex numbers $z_j$ satisfying the conditions $|z_j| = 1, z_j \not= 1$ for $j=1,..., n$ and for every $z_j$ there exists an $ i$ such that $z_i = \bar{z_j}. $…

Let $S_{\rm lcm}(n)$ denote the set of permutations $\pi$ of $[n]=\{1,2,\dots,n\}$ such that ${\rm lcm}[j,\pi(j)]\le n$ for each $j\in[n]$. Further, let $S_{\rm div}(n)$ denote the number of permutations $\pi$ of $[n]$ such that…

We prove a generalization of Frieman's $3k-3$ theorem for the sumset $$ \Sigma^{l}(A_1,\ldots,A_k)=\{a_{j_{1}}+\cdots+a_{j_{l}}:\,1\leq j_{1}<\cdots<j_{l}\leq k,\ a_{j_{s}}\in A_{j_{s}}\text{ for all }s\}. $$

This is a survey on the finite basis problem for varieties of algebraic systems. Our exposition is in two directions: (i) We give numerous examples of varieties which are not finitely based. (ii) We give examples of important varieties with…

Let $A \subset \mathbb{Z}^d$ be a finite set. It is known that $NA$ has a particular size ($\vert NA\vert = P_A(N)$ for some $P_A(X) \in \mathbb{Q}[X]$) and structure (all of the lattice points in a cone other than certain exceptional…

In this note, we give the explicit formula for the number of multisubsets of a finite abelian group $G$ with any given size such that the sum is equal to a given element $g\in G$. This also gives the number of partitions of $g$ into a given…

We show that if $\lambda_1,\ldots,\lambda_k$ are algebraic numbers, then $$|A+\lambda_1\cdot A+\dots+\lambda_k\cdot A|\geq H(\lambda_1,\ldots,\lambda_k)|A|-o(|A|)$$ for all finite subsets $A$ of $\mathbb{C}$, where…

Let $\Z/pZ$ be the finite field of prime order $p$ and $A$ be a subsequence of $\Z/pZ$. We prove several classification results about the following questions: (1) When can one represent zero as a sum of some elements of $A$ ? (2) When can…

Let $\alpha = (1+\sqrt{5})/2$ and define the lower and upper Wythoff sequences by $a_i = \lfloor i \alpha \rfloor$, $b_i = \lfloor i \alpha^2 \rfloor$ for $i \geq 1$. In a recent interesting paper, Kawsumarng et al. proved a number of…

This paper improves on a sum-product estimate obtained by Katz and Shen for subsets of a finite field whose order is not prime.

In this paper, we derive an explicit combinatorial formula for the number of $k$-subset sums of quadratic residues over finite fields.

In this note we continue the analysis of permutations that avoid substrings j(j+k), 1 <= j <= n-k, k < n, as well as substrings j(j+k) (mod n), 1 <= j <= n. In the first case the number of such permutations can be obtained from recursions…

A set of reals A={a_1,...,a_2} is called convex if a_{i+1} - a_i > a_i - a_{i-1} for all i. We prove, in particular, that |A-A| \gg |A|^{8/5} \log{-2/5} |A|.

A set $A\subset \mathbb{F}_p^n$ is sum-free if $A+A$ does not intersect $A$. If $p\equiv 2 \mod 3$, the maximal size of a sum-free in $\mathbb{F}_p^n$ is known to be $(p^n+p^{n-1})/3$. We show that if a sum-free set $A\subset…

Let $\mathcal{A}=(a_n)_{n\in\mathbb{N}_+}$ be a sequence of positive integers. Let $p_\mathcal{A}(n,k)$ denote the number of multi-color partitions of $n$ into parts in $\{a_1,\ldots,a_k\}$. We examine several arithmetic properties of the…