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

The kth finite subset space of a topological space X is the space exp_k X of non-empty finite subsets of X of size at most k, topologised as a quotient of X^k. The construction is a homotopy functor and may be regarded as a union of…

We study a variant of the Ackermann encoding $\mathbb{N}(x) := \sum_{y\in x}2^{\mathbb{N}(y)}$ of the hereditarily finite sets by the natural numbers, applicable to the larger collection $\mathsf{HF}^{1/2}$ of the hereditarily finite…

Let $A$ be a finite set of integers. We show that if $k$ is a prime power or a product of two distinct primes then $$|A+k\cdot A|\geq(k+1)|A|-\lceil k(k+2)/4\rceil$$ provided $|A|\geq (k-1)^{2}k!$, where $A+k\cdot A=\{a+kb:\ a,b\in A\}$. We…

The main result of this paper is the following: for all $b \in \mathbb Z$ there exists $k=k(b)$ such that \[ \max \{ |A^{(k)}|, |(A+u)^{(k)}| \} \geq |A|^b, \] for any finite $A \subset \mathbb Q$ and any non-zero $u \in \mathbb Q$. Here,…

Let $S$ and $T$ be subsets of $\mathbf{F}_q^n$. We show there are subsets $S'$ of $S$ and $T'$ of $T$ such that $S+T$ is the union of $S+T'$ and $S'+T$, with $|S'| + |T'|$ bounded by $c^n$ with $c < q$. The proof relies on the method of…

We bound double sums of Kloosterman sums over a finite field ${\mathbb F}_{q}$, with one or both parameters ranging over an affine space over its prime subfield ${\mathbb F}_p \subseteq {\mathbb F}_{q} $. These are finite fields analogues…

Let $K$ be a number field with ring of integers $\mathcal{O}_K$. Let $\mathcal{N}_K$ be the set of positive integers $n$ such that there exist units $\varepsilon, \delta \in \mathcal{O}_K^\times$ satisfying $\varepsilon + \delta = n$. We…

For any set $A$ of natural numbers with positive upper Banach density and any $k\geq 1$, we show the existence of an infinite set $B\subset{\mathbb N}$ and a shift $t\geq0$ such that $A-t$ contains all sums of $m$ distinct elements from $B$…

A More Sums Than Differences (MSTD, or sum-dominant) set is a finite set $A\subset \mathbb{Z}$ such that $|A+A|<|A-A|$. Though it was believed that the percentage of subsets of $\{0,...,n\}$ that are sum-dominant tends to zero, in 2006…

A set of sets is called a family. Two families $\mathcal{A}$ and $\mathcal{B}$ of sets are said to be cross-intersecting if each member of $\mathcal{A}$ intersects each member of $\mathcal{B}$. For any two integers $n$ and $k$ with $1 \leq…

R. Jin showed that whenever A and B are sets of integers having positive upper Banach density, the sumset A+B is piecewise syndetic. This result was strengthened by Bergelson, Furstenberg, and Weiss to conclude that A+B must be piecewise…

We show that a finite set of integers $A \subseteq \mathbb{Z}$ with $|A+A| \le K |A|$ contains a large piece $X \subseteq A$ with Fre\u{i}man dimension $O(\log K)$, where large means $|A|/|X| \ll \exp(O(\log^2 K))$. This can be thought of…

We announce two breakthrough results concerning important questions in the Theory of Computational Complexity. In this expository paper, a systematic and comprehensive geometric characterization of the Subset Sum Problem is presented. We…

Let $A\subseteq \mathbb{Z}_{\geq 0}$ be a finite set with minimum element $0$, maximum element $m$, and $\ell$ elements strictly in between. Write $(hA)^{(t)}$ for the set of integers that can be written in at least $t$ ways as a sum of $h$…

Let $n > k > 1$ be integers, $[n] = \{1, \ldots, n\}$. Let $\mathcal F$ be a family of $k$-subsets of~$[n]$. The family $\mathcal F$ is called intersecting if $F \cap F' \neq \emptyset$ for all $F, F' \in \mathcal F$. It is called almost…

Let $\mathbb{F}_q$ be a finite field of order $q$, where $q$ is large odd prime power. In this paper, we improve some recent results on the additive energy of the distance set, and on sumsets of the distance set due to Shparlinski (2016).…

The special case of the hypergeometric function $_{2}F_{1}$ represents the binomial series $(1+x)^{\alpha}=\sum_{n=0}^{\infty}(\:\alpha n\:)x^{n}$ that always converges when $|x|<1$. Convergence of the series at the endpoints, $x=\pm 1$,…

Erd\H{o}s showed that every set of $n$ positive integers contains a subset of size at least $n/(k+1)$ containing no solutions to $x_1 + \cdots + x_k = y$. We prove that the constant $1/(k+1)$ here is best possible by showing that if $(F_m)$…

This article contains a proof of the MDS conjecture for $k \leq 2p-2$. That is, that if $S$ is a set of vectors of ${\mathbb F}_q^k$ in which every subset of $S$ of size $k$ is a basis, where $q=p^h$, $p$ is prime and $q$ is not and $k \leq…

We show that there is a rational vector space $V$ such that, whenever $V$ is finitely coloured, there is an infinite set $X$ whose sumset $X+X$ is monochromatic. Our example is the rational vector space of dimension…