相关论文: On Sums of Generating Sets in (Z_2)^n

For $\lambda \in \mathbb{Z}$, let $\lambda \cdot A = \{ \lambda a : a \in A\}$. Suppose $r, h\in \mathbb{Z}$ are sufficiently large and comparable to each other. We prove that if $|A+A| \le K |A|$ and $\lambda_1, \ldots, \lambda_h \le 2^r$,…

Let $G$ be a graph on $n$ vertices and $(H,+)$ be an abelian group. What is the minimum size ${\sf S}_H(G)$ of the set of all sums $A(u)+A(v)$ over all injections $A:V(G)\to H$? In 2012, the first author, Angel, the second author, and…

Let $A_1,\ldots,A_n$ be finite subsets of an additive abelian group $G$ with $|A_1|=\cdots=|A_n|\ge2$. Concerning the two new kinds of restricted sumsets $$L(A_1,\ldots,A_n)=\{a_1+\cdots+a_n:\ a_1\in A_1,\ldots,a_n\in A_n,\ \text{and}\…

Given a compact convex domain $C\subset \mathbb{R}^k$ and bounded measurable functions $f_1,\ldots,f_n:C\to \mathbb{R}$, define the sup-convolution $(f_1\ast \ldots \ast f_n)(z)$ to be the supremum average value of…

We construct W-algebra generalizations of the ^sl(2) algebra -- W-algebras W^{(2)}_n generated by two currents E and F with the highest pole of order n in their OPE. The n=3 term in this series is the Bershadsky--Polyakov algebra. We define…

Let $\pi:{\mathbb R}^n \to {\mathbb R}^d$ be any linear projection, let $A$ be the image of the standard basis. Motivated by Postnikov's study of postitive Grassmannians via plabic graphs and Galashin's connection of plabic graphs to slices…

For positive integers $m$ and $n$, the Zarankiewicz number $Z_{2,2}(m,n)$ can be defined as the maximum total degree of a linear hypergraph with $m$ vertices and $n$ edges. Guy determined $Z_{2,2}(m,n)$ for all $n \geq \binom{m}{2}/3+O(m)$.…

In this paper we give a different approach to determining the cardinality of $h$-fold sumsets $hA$ when $A\subset \mathbb{Z}^d$ has $d+2$ elements. This enables us to provide more general result with a shorter and simpler proof. We also…

In this paper we present a procedure which allows to transform a subset $A$ of $\mathbb{Z}_{p}$ into a set $ A'$ such that $ |2\hspace{0.15cm}\widehat{} A'|\leq|2\hspace{0.15cm}\widehat{} A | $, where $2\hspace{0.15cm}\widehat{} A$ is…

For $p$ being a large prime number, and $A \subset \mathbb{F}_p$ we prove the following: $(i)$ If $A(A+A)$ does not cover all nonzero residues in $\mathbb{F}_p$, then $|A| < p/8 + o(p)$. $(ii)$ If $A$ is both sum-free and satisfies $A =…

Let M(n, d) be the maximum size of a permutation array on n symbols with pairwise Hamming distance at least d. We use various combinatorial, algebraic, and computational methods to improve lower bounds for M(n, d). We compute the Hamming…

Let $A$ be a nonempty finite subset of an additive abelian group $G$. Given a nonnegative integer $h$, the $h$-fold sumset $hA$ is the set of all sums of $h$ elements of $A$, and the restricted $h$-fold sumset $h^\wedge A$ is the set of all…

Let $A, B\subseteq \mathbb{R}^2$ be finite, nonempty subsets, let $s\geq 2$ be an integer, and let $h_1(A,B)$ denote the minimal number $t$ such that there exist $2t$ (not necessarily distinct) parallel lines,…

An unlabeled poset is said to be (2+2)-free if it does not contain an induced subposet that is isomorphic to 2+2, the union of two disjoint 2-element chains. Let $p_n$ denote the number of (2+2)-free posets of size $n$. In a recent paper,…

We consider a family, $\mathcal{F}$, of subsets of an $n$-set such that the cardinality of the symmetric difference of any two elements $F,F'\in\mathcal{F}$ is not a multiple of 4. We prove that the maximal size of $\mathcal{F}$ is bounded…

We prove that for sets $A, B, C \subset \mathbb{F}_p$ with $|A|=|B|=|C| \leq \sqrt{p}$ and a fixed $0 \neq d \in \mathbb{F}_p$ holds $$ \max(|AB|, |(A+d)C|) \gg|A|^{1+1/26}. $$ In particular, $$ |A(A+1)| \gg |A|^{1 + 1/26} $$ and $$…

Consider a set of $r$ convex $d$-polytopes $P_1,P_2,...,P_r$, where $d\ge{}3$ and $r\ge{}2$, and let $n_i$ be the number of vertices of $P_i$, $1\le{}i\le{}r$. It has been shown by Fukuda and Weibel that the number of $k$-faces of the…

Let $F_p$ be the field of a prime order $p.$ For a subset $A\subset F_p$ we consider the product set $A(A+1).$ This set is an image of $A\times A$ under the polynomial mapping $f(x,y)=xy+x:F_p\times F_p\to F_p.$ In the present paper we show…

Given an Abelian groups $G$, denote $\mu(G)$ the size of its largest sum-free subset and $f_{\max}(G)$ the number of maximal sum-free sets in $G$. Confirming a prediction by Liu and Sharifzadeh, we prove that all even-order $G\ne…

Ruzsa's conjecture asserts that any sequence $(a_n)_{n \geq 0}$ of integers that preserves congruences, $\textit{i.e.}$, satisfies $ a_{n+k} \equiv a_n \mod k $, and has the growth condition $\limsup_{n \to +\infty} |a_n|^{1/n} < e$, must…