Related papers: An asymptotically tight bound for the Davenport co…

Consider the geometric graph on $n$ independent uniform random points in a connected compact region $A$ of ${\bf R}^d, d \geq 2$, with $C^2$ boundary, or in the unit square, with distance parameter $r_n$. Let $K_n$ be the number of…

For each $N\geq 1$, let $G_N$ be a simple random graph on the set of vertices $[N]=\{1,2, ..., N\}$, which is invariant by relabeling of the vertices. The asymptotic behavior as $N$ goes to infinity of correlation functions: $$ \mathfrak…

A generalization of the Davenport constant is investigated. For a finite abelian group $G$ and a positive integer $k$, let $D_k(G)$ denote the smallest $\ell$ such that each sequence over $G$ of length at least $\ell$ has $k$ disjoint…

Let $\mathcal{S}$ be a finite commutative semigroup. The Davenport constant of $\mathcal{S}$, denoted ${\rm D}(\mathcal{S})$, is defined to be the least positive integer $\ell$ such that every sequence $T$ of elements in $\mathcal{S}$ of…

We prove an asymptotic Cram\'er's theorem, that is, if the sequence $(X_{n}+ Y_{n})_{n\geq 1}$ converges in law to the standard normal distribution and for every $n\geq 1$ the random variables $X_{n}$ and $Y_{n}$ are independent, then…

An integer $n\ge 1$ is said to be practical if every natural number $ m \le n$ can be expressed as a sum of distinct positive divisors of $n$. The number of practical numbers up to $x$ is asymptotic to $c x/\log x$, where $c$ is a constant.…

We prove that there exists an entire function for which every complex number is an asymptotic value and whose growth is arbitrarily slow subject only to the necessary condition that the function is of infinite order.

The conjectured Robin inequality for an integer $n>7!$ is $\sigma(n)<e^\gamma n \log \log n,$ where $\gamma$ denotes Euler constant, and $\sigma(n)=\sum_{d | n} d $. Robin proved that this conjecture is equivalent to Riemann hypothesis…

The isotropy constant of any $d$-dimensional polytope with $n$ vertices is bounded by $C \sqrt{n/d}$ where $C>0$ is a numerical constant.

W.M.Schmit[11] conjectured that for any$\;\theta$ with deg$\;\theta\geq 3,$ there is no constant$\;C=C(\theta)$ so that$\;|p-q\theta|>Cq^{-1}$ for every rationa$\;p/q.$ [12,p26] states that the computations of the first several thousand…

We establish sharp estimates for the convergence rate of the Kranosel'ski\v{\i}-Mann fixed point iteration in general normed spaces, and we use them to show that the asymptotic regularity bound recently proved in [11] (Israel Journal of…

I present an asymptotic formula for the Takeuchi numbers $T_n$. In particular, I give compelling numerical evidence and present a heuristic argument showing that $$T_n\sim C_T B_n\exp{1\over2}{W(n)}^2$$as $n$ tends to infinity, where $B_n$…

We prove that the pigeonhole upper bound $\lambda(s,m) \leq \binom{m}{2}(s+1)$ is asymptotically tight whenever $s/\!\sqrt{m} \to \infty$. In particular, $\lambda(s,m) \sim \binom{m}{2}\,s$ in this regime. As corollaries: $\lambda(n,n)/n^3…

Let X:R2\Dr->R2 be a differentiable (but not necessarily C1) vector field, where r>0 and Dr={z\in R2:|z|\le r}. If for some e>0 and for all p\in R2\Dr, no eigenvalue of D_p X belongs to (-e,0]\cup {z\in\C:\mathcal{R}(z)\ge 0}, then (a)For…

We prove almost sure strong asymptotic freeness of i.i.d. random unitaries with the following law: sample a Haar unitary matrix of dimension $n$ and then send this unitary into an irreducible representation of $U(n)$. The strong convergence…

Let $G$ be a finite group. By a sequence over $G$, we mean a finite unordered string of terms from $G$ with repetition allowed, and we say that it is a product-one sequence if its terms can be ordered so that their product is the identity…

Using the sum invariance property of Benford random variables, we prove that an $n$-digit Benford variable converges to a Benford variable as $n$ approaches infinity.

It is known that the asymptotic density of states of a 2d CFT in an irreducible representation $\rho$ of a finite symmetry group $G$ is proportional to $(\dim\rho)^2$. We show how this statement can be generalized when the symmetry can be…

Let \sigma(n) = \sum_{d \mid n}d be the usual sum-of-divisors function. In 1933, Davenport showed that that n/\sigma(n) possesses a continuous distribution function. In other words, the limit D(u):= \lim_{x\to\infty} \frac{1}{x}\sum_{n \leq…

Given a finite commutative semigroup $\mathcal{S}$ (written additively), denoted by ${\rm D}(\mathcal{S})$ the Davenport constant of $\mathcal{S}$, namely the least positive integer $\ell$ such that for any $\ell$ elements…