Related papers: On large Sidon sets

For every pattern $P$, consisting of a finite set of points in the plane, $S_{P}(n,m)$ is defined as the largest number of similar copies of $P$ among sets of $n$ points in the plane without $m$ points on a line. A general construction,…

Let $F\in\mathbb{Z}[x,y]$ and $m\ge2$ be an integer. A set $A\subset \mathbb{Z}$ is called an $(F,m)$-Diophantine set if $F(a,b)$ is a perfect $m$-power for any $a,b\in A$ where $a\ne b$. If $F$ is a bivariate polynomial for which there…

Low correlation (finite length) sequences are used in communications and remote sensing. One seeks codebooks of sequences in which each sequence has low aperiodic autocorrelation at all nonzero shifts, and each pair of distinct sequences…

To each finitely generated group $G$, we associate a quasi-isometric invariant called the \emph{Dehn spectrum} of $G$. If $G$ is finitely presented, our invariant is closely related to the Dehn function of $G$, but provides more information…

We solve an elementary extremal problem on trigonometric polynomials and obtain the exact value of the Sidon constant for sets with three elements {n0, n1, n2}: it is \sec(\pi / 2n) for n = max(n1-n0, n2-n0, n2-n1) / gcd(n1-n0, n2-n0).

We prove the existence of clopen marker sets with some strong regularity property. For each $n\geq 1$ and any integer $d\geq 1$, we show that there are a positive integer $D$ and a clopen marker set $M$ in $F(2^{\mathbb{Z}^n})$ such that…

The purpose of this short problem paper is to raise an extremal question on set systems which seems to be natural and appealing. Our question is: which set systems of a given size maximise the number of $(n+1)$-element chains in the power…

A finite subset $X$ of the Euclidean space is called an $m$-distance set if the number of distances between two distinct points in $X$ is equal to $m$. An $m$-distance set $X$ is said to be maximal if any vector cannot be added to $X$ while…

Recently, there has been interest in representing single graphs by multiple drawings; for example, using graph stories, storyplans, or uncrossed collections. In this paper, we apply this idea to orthogonal graph drawing. Due to the…

We show that the set of realizations of a given dimension of a max-plus linear sequence is a finite union of polyhedral sets, which can be computed from any realization of the sequence. This yields an (expensive) algorithm to solve the…

We construct large Salem sets avoiding patterns, complementing previous constructions of pattern avoiding sets with large Hausdorff dimension. For a (possibly uncountable) family of uniformly Lipschitz functions $\{ f_i :…

Consider families of $k$-subsets (or blocks) on a ground set of size $v$. Recall that if all $t$-subsets occur with the same frequency $\lambda$, one obtains a $t$-design with index $\lambda$. On the other hand, if all $t$-subsets occur…

Let $k \ge 2$ be an integer. We say a set $A$ of positive integers is an asymptotic basis of order $k$ if every large enough positive integer can be represented as the sum of $k$ terms from $A$. A set of positive integers $A$ is called…

A set $S\subset \mathbb{N}$ is a Sidon set if all pairwise sums $s_1+s_2$ (for $s_1, s_2\in S$, $s_1\leq s_2$) are distinct. A set $S\subset \mathbb{N}$ is an asymptotic basis of order 3 if every sufficiently large integer $n$ can be…

We use semidefinite programming to bound the fractional cut-cover parameter of graphs in association schemes in terms of their smallest eigenvalue. We also extend the equality cases of a primal-dual inequality involving the…

A point set $M$ in Euclidean plane is called an integral point set in semi-general position if all the distances between the elements of $M$ are integers, and $M$ does not contain collinear triples. We improve the lower bound for diameter…

Generalising results of Erd\H{o}s-Freud and Lindstr\"om, we prove that the largest Sidon subset of a bounded interval of integers is equidistributed in Bohr neighbourhoods. We establish this by showing that extremal Sidon sets are…

Let X be a subset of an abelian group and a_1,...,a_h,a'_1,...,a'_h a sequence of 2h elements of X such that a_1 + ... + a_h = a'_1 + ... + a'_h. The set X is a Sidon set of order h if, after renumbering, a_i = a'_i for i = 1,..., h. For k…

Let $L=(L_d)_{d \in \mathbb N}$ be any ordered probability sequence, i.e., satisfying $0 < L_{d+1} \le L_d$ for each $d \in \mathbb N$ and $\sum_{d \in \mathbb N} L_d =1$. We construct sequences $A = (a_i)_{i \in \mathbb N}$ on the…

Domination of grids has been proved to be a demanding task and with the addition of independence it becomes more challenging. It is known that no grid with $m,n \geq 5$ has an efficient dominating set, also called perfect code, that is, an…