Related papers: A note on the Duffin-Schaeffer conjecture

In this note, in particular, we establish the following result: Let $X$ be a real Banach space, $\varphi\in X^*\setminus \{0\}$ and $\psi:X\to {\bf R}$ a Lipschitzian functional with Lipschitz constant equal to $\varphi\|_X^{*}$. Then, we…

Let $x\ge 2$. The $\psi$-form of the prime number theorem is $\psi(x) =\sum\sb{n \le x}\Lambda(n) =x +O\bigl(x\sp{1-H(x)} \log\sp{2} x\big)$, where $H(x)$ is a certain function of $x$ with $0< H(x) \le \tfrac{1}{2}$. Tur\'an proved in 1950…

We show that the de Bruijn-Erd\H{o}s condition for the error term in their improvement of Fekete's Lemma is not only sufficient but also necessary in the following strong sense. Suppose that given a sequence $0\leq f(1)\leq f(2)\leq…

Let $G$ denotes a finite abelian group of order $n$ and Davenport constant $D$, and put $m= n+D-1$. Let $x=(x_1, ..., x_m)\in G^m$ be a sequence with a maximal repetition $\ell$ attained by $x_m$ and put $r=\min(D,\ell)$. Let $w=(w_1, ...,…

Let S $\subseteq$ N be a numerical semigroup with multiplicity m, conductor c and minimal generating set P. Let L = S $\cap$ [0, c -- 1] and W(S) = |P||L| -- c. In 1978, Herbert Wilf asked whether W(S) $\ge$ 0 always holds, a question known…

We develop a metric and probabilistic theory for the Ostrogradsky representation of real numbers, i.e., the expansion of a real number $x$ in the following form: \begin{align*} x&= \sum_n\frac{(-1)^{n-1}}{q_1q_2... q_n}=…

Let us draw a graph R on {0,1,...,n-1} by having an edge {i,j} with probability p_(|i-j|), where sum_i p_i is finite and let M_n=(n,<,R). For a first order sentence psi let a^n_psi be the probability of ``M_n satisfies psi''. We prove that…

For a non-negative integer $m$, let $S(m)$ denote the sum given by $$S(m):=\sum_{n=0}^{m}\frac{(-1)^n(8n+1)}{n!^3}\left(\frac{1}{4}\right)_n^3.$$ Using the powerful WZ-method, for a prime $p\equiv 3$ $($mod $4)$ and an odd integer $r>1$, we…

For $n \in \mathbb{N}$ let $\Pi[n]$ denote the set of partitions of $n$, i.e., the set of positive integer tuples $(x_1,x_2,\ldots,x_k)$ such that $x_1 \geq x_2 \geq \cdots \geq x_k$ and $x_1 + x_2 + \cdots + x_k = n$. Fixing…

Fix $p\in[1,\infty)$, $K\in(0,\infty)$ and a probability measure $\mu$. We prove that for every $n\in\mathbb{N}$, $\varepsilon\in(0,1)$ and $x_1,\ldots,x_n\in L_p(\mu)$ with $\big\| \max_{i\in\{1,\ldots,n\}} |x_i| \big\|_{L_p(\mu)} \leq K$,…

Let $ (\bx(n))_{n \geq 1} $ be an $s-$dimensional Niederreiter-Xing sequence in base $b$. Let $D((\bx(n))_{n = 1}^{N})$ be the discrepancy of the sequence $ (\bx(n))_{n = 1}^{N} $. It is known that $N D((\bx(n))_{n = 1}^{N}) =O(\ln^s N)$ as…

Let $G$ be a finite group and let $\psi(G)$ denote the sum of element orders of $G$. It is well-known that the maximum value of $\varphi$ on the set of groups of order $n$, where $n$ is a positive integer, will occur at the cyclic group…

Let $Q$ be an infinite subset of $\mathbb{Z}$, let $\Psi: \mathbb{Z} \rightarrow [0,\infty)$ be positive on $Q$, and let $\theta \in \mathbb{R}$. Define $$ E(Q,\Psi,\theta) = \{ x \in \mathbb{R} : \| q x - \theta \| \leq \Psi(q) \text{ for…

G. K. Pedersen and M. Takesaki have proved in 1973 that if $\varphi$ is a faithful, semi-finite, normal weight on a von Neumann algebra $M\;\!$, and $\psi$ is a $\sigma^{\varphi}$-invariant, semi-finite, normal weight on $M\;\!$, equal to…

This paper takes a new step in the direction of proving the Duffin-Schaeffer Conjecture for measures arbitrarily close to Lebesgue. The main result is that under a mild `extra divergence' hypothesis, the conjecture is true.

Let $\mathcal{W}$ be the set of strongly real elements of $W$, a Coxeter group. Then for $w \in \mathcal{W}$, $e(w)$, the excess of $w$, is defined by $e(w) = \min\{\ell(x) + \ell(y) - \ell(w) \; | \; w=xy, x^2 = y^2 = 1\}$. When $W$ is…

We approximate intersection numbers $\big\langle \psi_1^{d_1}\cdots \psi_n^{d_n}\big\rangle_{g,n}$ on Deligne-Mumford's moduli space $\overline{\mathcal M}_{g,n}$ of genus $g$ stable complex curves with $n$ marked points by certain…

The inhomogeneous Khintchine-Groshev Theorem is a classical generalization of Khintchine's Theorem in Diophantine approximation, by approximating points in $\mathbb{R}^m$ by systems of linear forms in $n$ variables. Analogous to the…

Denote the sum of element orders in a finite group $G$ by $\psi(G)$ and let $C_n$ denote the cyclic group of order $n$. In this paper, we prove that if $|G|=n$ and $\psi(G)>\frac{13}{21}\,\psi(C_n)$, then $G$ is nilpotent. Moreover, we have…

A (d-parameter) basic nilsequence is a sequence of the form \psi(n)=f(a^{n}x), n \in Z^{d}, where x is a point of a compact nilmanifold X, a is a translation on X, and f is a continuous function on X; a nilsequence is a uniform limit of…