Related papers: $B_h$-sets of real and complex numbers

We show that every set $A$ of natural numbers with positive upper density can be shifted to contain the restricted sumset $\{b_1 + b_2 : b_1, b_2\in B \text{ and } b_1 \neq b_2 \}$ for some infinite set $B \subset A$.

Given a Fell bundle $\mathcal{B}=\{B_t\}_{t\in G}$ over a LCH group and a closed subgroup $H\subset G,$ we show that all the *-representations of $\mathcal{B}_H:=\{B_t\}_{t\in H}$ can be induced to *-representations of $\mathcal{B}$ by…

For $g \geq 2$ and $h \geq 3$, we give small improvements on the maximum size of a $B_h[g]$-set contained in the interval $\{1,2, \dots , N \}$. In particular, we show that a $B_3[g]$-set in $\{1,2, \dots , N \}$ has at most $(14.3 g…

Given a connected graph $G$ on $n$ vertices and a positive integer $k\le n$, a subgraph of $G$ on $k$ vertices is called a $k$-subgraph in $G$. We design combinatorial approximation algorithms for finding a connected $k$-subgraph in $G$…

An important theorem of geometric measure theory (first proved by Besicovitch and Davies for Euclidean space) says that every analytic set of non-zero $s$-dimensional Hausdorff measure $\mathcal H^s$ contains a closed subset of non-zero…

In this article, we first describe all nonempty sets of integers S with the property that for all n and m in S, not necessarily distinct, the set {n-m,n+m} intersected with S consists of a single element. These are the sets with at most two…

Absolutely representing system (ARS) in a Banach space $X$ is a set $D \subset X$ such that every vector $x$ in $X$ admits a representation by an absolutely convergent series $x = \sum_i a_i x_i$ with $(a_i)$ reals and $(x_i) \subset D$. We…

The Helton-Nie Conjecture (HNC) is the proposition that every convex semialgebraic set is a spectrahedral shadow. Here we prove that HNC is equivalent to another propo- sition related to quadratically constrained quadratic programming.…

Let X be a smooth algebraic curve of genus g>=2. A stable vector bundle over X of degree d, rank n with at least k sections is called a Brill-Noether bundle of type (n,d,k). By tensoring coherent systems, we prove that most of the known…

Let $G$ be a finite abelian group and let $\exp(G)$ denote the least common multiple of the orders of all elements of $G$. A $BH(G,h)$ matrix is a $G$-invariant $|G|\times |G|$ matrix $H$ whose entries are complex $h$th roots of unity such…

We analyze sumsets A+B = {a+b : a in A, b in B} where A,B are sets of integers, A is infinite, and B has positive upper Banach density. For each k, we show that A+B contains at least the expected density of k-term arithmetic progressions…

Large sets of combinatorial designs has always been a fascinating topic in design theory. These designs form a partition of the whole space into combinatorial designs with the same parameters. In particular, a large set of block designs,…

Let $G$ be a linear algebraic group over an infinite field $k$. Loosely speaking, a $G$-torsor over $k$-variety is said to be versal if it specializes to every $G$-torsor over any $k$-field. The existence of versal torsors is well-known. We…

For an element $a$ of a monoid $H$, its set of lengths $\mathsf L (a) \subset \mathbb N$ is the set of all positive integers $k$ for which there is a factorization $a=u_1 \cdot \ldots \cdot u_k$ into $k$ atoms. We study the system $\mathcal…

Consider the following classes of pairs consisting of a group and a finite collection of subgroups: $\mathcal{C}= \left\{ (G,\mathcal H) \mid \text{$\mathcal{H}$ is hyperbolically embedded in $G$} \right\}$ and $ \mathcal{D}= \left\{…

Let $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, and $T_n(\mathbb{K})$ be the set of $n\times n$ lower triangular matrices with entries in $\mathbb{K}$. We show that $T_n(\mathbb{K})$ has dense subsemigroups that are generated by $n+1$…

We show that if $A=\{a_1 < a_2 < \ldots < a_k\}$ is a set of real numbers such that the differences of the consecutive elements are distinct, then for and finite $B \subset \mathbb{R}$, $$|A+B|\gg |A|^{1/2}|B|.$$ The bound is tight up to…

New sets (typically found by computer search) with Sidon constant equal to the square root of their cardinalities are given. For each integer $N$ there are only a finite number of groups of prime order containing $N$-element extreme sets.…

We prove that if a subset of the d-dimensional vector space over a finite field is large enough, then it contains many k-tuples of mutually orthogonal vectors.

We study the representations of the commutator subgroup K_{n} of the braid group B_{n} into a finite group . This is done through a symbolic dynamical system. Some experimental results enable us to compute the number of subgroups of K_{n}…