Related papers: More differences than multiple sums

We construct a convex set $A$ with cardinality $2n$ and with the property that an element of the difference set $A-A$ can be represented in $n$ different ways. We also show that this construction is optimal by proving that for any convex…

It is established that for any finite set of positive real numbers $A$, we have $$|A/A+A| \gg \frac{|A|^{\frac{3}{2}+\frac{1}{26}}}{\log^{1/2}|A|}.$$

Let $A$ be a finite set of positive real numbers. We present a sum-division estimate: |A+A|^2|A/A|\geq\frac{|A|^4}{4}.

In this note we associate a sequence of non-negative integers to any convergent series of positive real numbers and study this sequence for the series $\sum_{n \geq 1} n^{-k}$ where $k$ is an integer $\geq 2$.

A sum where each of the $N$ summands can be independently chosen from two choices yields $2^N$ possible summation outcomes. There is an $\mathcal{O}(K^2)$-algorithm that finds the $K$ smallest/largest of these sums by evading the…

Two families $\mathcal A$ and $\mathcal B$ of $k$-subsets of an $n$-set are called cross-intersecting if $A\cap B\ne\emptyset$ for all $A\in \mathcal A, B\in \mathcal B $. Strengthening the classical Erd\H os-Ko-Rado theorem, Pyber proved…

We study fundamental groups of compact Sasaki manifolds and show that compared to K\"ahler groups, they exhibit rather different behaviour. This class of groups is not closed under taking direct products, and there is often an upper bound…

Let $A$ and $B$ be sets of $k\ge5$ elements in $F=\mathbb{Z}/p\mathbb{Z}$ the field with $p>2k-2$ elements. We denote by $A\dot{+}B$ the set of different elements of $F$ that can be written in the form $a+b$, where $a\in A$, $b\in B$,…

Let $\tau_k(n)$ be the $k$-th divisor function. In this paper, we derive an asymptotic formula for the sum $$ \sum_{1\leq n_1,n_2, \dots, n_{\ell}\leq X^{\frac{1}{r}} \atop 1\leq n_{\ell+1}\le X^{\frac{1}{s}}}\tau_k(n_1^r+n_2^r+\dots…

Let $A, B\subseteq \mathbb{Z}$ be finite, nonempty subsets with $\min A=\min B=0$, and let $$\delta(A,B)={\begin{array}{ll} 1 & \hbox{if} A\subseteq B, 0 & \hbox{otherwise.} If $\max B\leq \max A\leq |A|+|B|-3$ and \label{one}|A+B|\leq…

We show that if $A$ is a subset of a group of prime order $p$ such that $|2A|<2.7652|A|$ and $|A|<1.25\cdot10^{-6}p$, then $A$ is contained in an arithmetic progression with at most $|2A|-|A|+1$ terms, and $2A$ contains an arithmetic…

Let $k \geq 1$ be an integer. A set $A \subset \mathbb{Z}$ is a $k$-fold Sidon set if $A$ has only trivial solutions to each equation of the form $c_1 x_1 + c_2 x_2 + c_3 x_3 + c_4 x_4 = 0$ where $0 \leq |c_i | \leq k$, and $c_1 + c_2 + c_3…

Nathanson introduced the range of cardinalities of $h$-fold sumsets $R(h,k) := \{|hA|:A \subset \mathbb{Z} \text{ and }|A| = k\}.$ Following a remark of Erd\H{o}s and Szemer\'edi that determined the form of $R(h,k)$ when $h=2$, Nathanson…

A finite set of integers $A$ is a sum-dominant (also called an More Sums Than Differences or MSTD) set if $|A+A| > |A-A|$. While almost all subsets of $\{0, \dots, n\}$ are not sum-dominant, interestingly a small positive percentage are. We…

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 $0 \leq k \leq n$, let ${[n] \choose \leq…

This is a sequel to the paper arXiv:1312.6438 by the same authors. In this sequel, we quantitatively improve several of the main results of arXiv:1312.6438, and build on the methods therein. The main new results is that, for any finite set…

Let $n=2k+1$ be odd with $k\geq3$. In this note, we give two intersecting families with diversity larger than $\sum_{i=k+1}^{2k} \binom {2k}{i}$, which disprove a conjecture of Huang.

Given a finite abelian group $G$, consider the complete graph on the set of all elements of $G$. Find a Hamiltonian cycle in this graph and for each pair of consecutive vertices along the cycle compute their sum. What are the smallest and…

A subset $A$ of $[n] = \{1, \dots, n\}$ is $k$-separated if, when the elements of $[n]$ are considered on a circle, between any two elements of $A$ there are at least $k$ elements of $[n]$ that are not in $A$. A family $\mathcal{A}$ of sets…

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|.