Related papers: On Sidon sets in a random set of vectors

A SIC is a maximal equiangular tight frame in a finite dimensional Hilbert space. Given a SIC in dimension $d$, there is good evidence that there always exists an aligned SIC in dimension $d(d-2)$, having predictable symmetries and smaller…

Let Sym_n denote the symmetric group of all permutations pi = a_1...a_n of {1,...,n}. An index i is a peak of pi if a_{i-1} < a_i > a_{i+1} and we let P(pi) be the set of peaks of pi. Given any set S of positive integers we define P(S;n) to…

Since ancient times mathematicians consider geometrical objects with integral side lengths. We consider plane integral point sets $\mathcal{P}$, which are sets of $n$ points in the plane with pairwise integral distances where not all the…

Is there a joint distribution of $n$ random variables over the natural numbers, such that they always form an increasing sequence and whenever you take two subsets of the set of random variables of the same cardinality, their distribution…

Motivated by the problem of computing the distribution of the largest distance $d_{\max}$ between $n$ random points on a circle we derive an explicit formula for the moments of the maximal component of a random vector following a Dirichlet…

Let $\omega$ be a sequence of positive integers. Given a positive integer $n$, we define $$ r_n(\omega) = | \{ (a,b)\in \mathbb{N}\times \mathbb{N}\colon a,b \in \omega, a+b = n, 0 <a<b \}|. $$ S. Sidon conjectured that there exists a…

Let $d$ be a positive integer. A finite group is called $d$-maximal if it can be generated by precisely $d$ elements, while its proper subgroups have smaller generating sets. For $d\in\{1,2\}$, the $d$-maximal groups have been classified up…

Let $S$ be a finite set, and $X_1,\ldots,X_n$ an i.i.d. uniform sample from $S$. To estimate the size $|S|$, without further structure, one can wait for repeats and use the birthday problem. This requires a sample size of the order…

Let $w=(w_1,\dots,w_d)$ be a $d$-tuple of positive real numbers such that $\sum_{i}w_i =1$ and $w_1\geq \cdots \geq w_d$. A $d$-dimensional vector $x=(x_1,\dots,x_d)\in\mathbb{R}^d$ is said to be $w$-singular if for every $\epsilon>0$ there…

We consider the binomial random graph $G(n,p)$, where $p$ is a constant, and answer the following two questions. First, given $e(k)=p{k\choose 2}+O(k)$, what is the maximum $k$ such that a.a.s.~the binomial random graph $G(n,p)$ has an…

Given a set of independent Poisson random variables with common mean, we study the distribution of their maximum and obtain an accurate asymptotic formula to locate the most probable value of the maximum. We verify our analytic results with…

A $d$-hypertree on $[n]$ is a maximal acyclic $d$-dimensional simplicial complex with full $(d-1)$-skeleton on the vertex set $[n]$. Alternatively, in the language of algebraic topology, it is a minimal $d$-dimensional simplicial complex…

Fix $\alpha >0$, and sample $N$ integers uniformly at random from $\{1,2,\ldots ,\lfloor e^{\alpha N}\rfloor \}$. Given $\eta >0$, the probability that the maximum of the pairwise GCDs lies between $N^{2-\eta }$ and $N^{2+\eta}$ converges…

The Corners Theorem states that for any $\alpha > 0$ there exists an $N_0$ such that for any abelian group $G$ with $|G| = N \geq N_0$ and any subset $A \subset G \times G$ with $|A| \ge \alpha N^2$ we can find a corner in $A$ , i.e. there…

We show that the statement ``In every separable pseudometric space there is a maximal non-strictly \delta-separated set.'' implies the axiom of choice for countable families of sets. This gives answers to a question of Dybowski and…

A set of integers is \emph{primitive} if it does not contain an element dividing another. Denote by $f(n)$ the number of maximum-size primitive subsets of $\{1,\ldots, 2n\}$. We prove that the limit $\alpha=\lim_{n\rightarrow…

The authors have recently obtained a lower bound of the Hausdorff dimension of the sets of vectors $(x_1, \ldots, x_d)\in [0,1)^d$ with large Weyl sums, namely of vectors for which $$ \left| \sum_{n=1}^{N}\exp(2\pi i (x_1 n+\ldots +x_d…

One of the prominent open problems in combinatorics is the discrepancy of set systems where each element lies in at most $t$ sets. The Beck-Fiala conjecture suggests that the right bound is $O(\sqrt{t})$, but for three decades the only…

The main result of this thesis is to show that there are only finitely many integers $n$ such that both $n$ and $d(n)$ are highly composite numbers at the same time, where $d(n)$ is the divisor function. Bertrand's postulate [4] is used…

For $\varepsilon\in(0,1/2)$ and a natural number $d\ge 2$, let $N$ be a natural number with \[ N \,\ge\, 2^9\,\log_2(d)\, \left(\frac{\log_2(1/\varepsilon)}{\varepsilon}\right)^2. \] We prove that there is a set of $N$ points in the unit…