Related papers: On sumsets in ${\Bbb F}_2^n$

For $B \subseteq \mathbb F_q^m$, the $n$-th affine extremal number of $B$ is the maximum cardinality of a set $A \subseteq \mathbb F_q^n$ with no subset which is affinely isomorphic to $B$. Furstenberg and Katznelson proved that for any $B…

Two subsets $A,B$ of an $n$-element ground set $X$ are said to be \emph{crossing}, if none of the four sets $A\cap B$, $A\setminus B$, $B\setminus A$ and $X\setminus(A\cup B)$ are empty. It was conjectured by Karzanov and Lomonosov forty…

The $h$-fold sumset of a set $A$ of integers is the set of all sums of $h$ not necessarily distinct elements of $A$. Let $(A_q)_{q=1}^{\infty}$ be a strictly decreasing sequence of sets of integers and let $A = \bigcap_{q=1}^{\infty} A_q$.…

Inspired by Karr's algorithm, we consider the summations involving a sequence satisfying a recurrence of order two. The structure of such summations provides an algebraic framework for solving the difference equations of form…

We prove the following theorem: for all positive integers $b$ there exists a positive integer $k$, such that for every finite set $A$ of integers with cardinality $|A| > 1$, we have either $$ |A + ... + A| \geq |A|^b$$ or $$ |A \cdot ...…

The article addresses the problem whether indefinite double sums involving a generic sequence can be simplified in terms of indefinite single sums. Depending on the structure of the double sum, the proposed summation machinery may provide…

For positive integers $n>k>t$ let $\binom{[n]}{k}$ denote the collection of all $k$-subsets of the standard $n$-element set $[n]=\{1,\ldots,n\}$. Subsets of $\binom{[n]}{k}$ are called $k$-graphs. A $k$-graph $\mathcal{F}$ is called…

Let $K_{q^n}(a)$ be a Kloosterman sum over the finite field $\F_{q^n}$ of characteristic $p$. In this note so called subfield conjecture is proved in case $p>3$: if $a\ne0$ belongs to the proper subfield $\F_q$ of $\F_{q^n}$, then…

For a finite vector space $V$ and a non-negative integer $r\le\dim V$ we estimate the smallest possible size of a subset of $V$, containing a translate of every $r$-dimensional subspace. In particular, we show that if $K\subset V$ is the…

We prove a conjecture that arose in the context of a subspace enumeration problem over finite fields. We prove, more generally, a bibasic, double-sum identity, which extends a $q$-analogue of the (terminating) binomial theorem.

In 1974, Erd\H{o}s and Kleitman conjectured that if a family $\mathcal{F}\subseteq 2^{[n]}$ contains no matching of size \(s\) and is maximal with respect to this property, then $ |\mathcal{F}|\ge \left(1-2^{-(s-1)}\right)\cdot 2^{n}. $ For…

We construct sets $A, B$ in a vector space over $\mathbb{F}_2$ with the property that $A$ is "statistically" almost closed under addition by $B$ in the sense that $a + b$ almost always lies in $A$ when $a \in A, b \in B$, but which is…

Let $A$ be a nonempty finite set of $k$ integers. Given a subset $B$ of $A$, the sum of all elements of $B$, denoted by $s(B)$, is called the subset sum of $B$. For a nonnegative integer $\alpha$ ($\leq k$), let \[\Sigma_{\alpha}…

We prove that for a continuum $K\subset \mathbb R^n$ the sum $K^{+n}$ of $n$ copies of $K$ has non-empty interior in $\mathbb R^n$ if and only if $K$ is not flat in the sense that the affine hull of $K$ coincides with $\mathbb R^n$.…

Let $d\geq 2$ and $k\geq 1$ be fixed. We prove that, for every $\epsilon>0$ and every real $\beta$, there exist integers $1\leq b_1,\ldots,b_k\leq N$ such that \[ \left\|\sum_{j=1}^k b_j^{1/d}-\beta\right\| \ll_{d,k,\epsilon}…

In this paper the authors study set expansion in finite fields. Fourier analytic proofs are given for several results recently obtained by Solymosi, Vinh and Vu using spectral graph theory. In addition, several generalizations of these…

Let A and B be subsets of an elementary abelian 2-group G, none of which are contained in a coset of a proper subgroup. Extending onto potentially distinct summands a result of Hennecart and Plagne, we show that if |A+B|<|A|+|B|, then…

For any set $A$ of natural numbers with positive upper Banach density, we show the existence of an infinite set $B$ and sequences $(t_k)_{k\in \mathbb{N}}, (s_k)_{k\in \mathbb{N}}$ of natural numbers such that $\left\{ \sum_{n \in F}n : F…

In this paper, we develop Terence Tao's harmonic analysis method and apply it to restricted sumsets. The well known Cauchy-Davenport theorem asserts that if $A$ and $B$ are nonempty subsets of $Z/pZ$ with $p$ a prime, then $|A+B|\ge…

Let $\omega(\mathcal{F})=\sum_{\{A,B\}\subset\mathcal{F}}|A\cap B|$ and $\omega(\mathcal{A},\mathcal{B})=\sum_{(A,B)\in \mathcal{A}\times \mathcal{B}}|A\cap B|$. A family $\mathcal{F}$ is intersecting if $F_1\cap F_2\neq \emptyset$ for any…