Related papers: A Collatz-type conjecture on the set of rational n…

We consider the partial theta function $\theta (q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$, where $q\in (-1,0)\cup (0,1)$ and either $x\in \mathbb{R}$ or $x\in \mathbb{C}$. We prove that for $x\in \mathbb{R}$, in each of the two cases $q\in…

We investigate the structure of $\omega$-limit (resp. $\alpha$-limit) sets for a monotone map $f$ on a regular curve $X$. %Let $X$ be a regular curve and let $f: X\longrightarrowX$ be a monotone map. We show that for any $x\in X$ (resp. for…

Let $f(z)=z^5+az^3+bz^2+cz+d \in \Z[z]$ and let us consider a del Pezzo surface of degree one given by the equation $\cal{E}_{f}: x^2-y^3-f(z)=0$. In this note we prove that if the set of rational points on the curve $E_{a,…

We show that there exists a positive arithmetical formula $\psi(x,R)$, where $x \in \omega$, $R \subseteq \omega$, with no hyperarithmetical fixed point. This answers a question of Gerhard J\"{a}ger. As corollaries we obtain results on the…

Let $\mathcal X$ be an infinite locally compact separable metric space with metric $\rho$ and let $f : \mathcal X \longrightarrow \mathcal X$ be a continuous weakly mixing map. Let $\beta = \sup \big\{ \rho(x, y): \{x, y \} \subset \mathcal…

When $A$ and $B$ are subsets of the integers in $[1,X]$ and $[1,Y]$ respectively, with $|A| \geq \alpha X$ and $|B| \geq \beta X$, we show that the number of rational numbers expressible as $a/b$ with $(a,b)$ in $A \times B$ is $\gg (\alpha…

Given a natural number $k\ge 2$ and a $k$-automatic set $S$ of natural numbers, we show that the lower density and upper density of $S$ are recursively computable rational numbers and we provide an algorithm for computing these quantities.…

The 3x+1 semigroup is the multiplicative semigroup generated by the rational numbers of form (2k+1)/(3k+2) for non-negative k, together with 2. This semigroup encodes backward iteration under the 3x+1 map, and the 3x+1 conjecture implies…

Let X be a Noetherian space, let f be a continuous self-map on X, let Y be a closed subset of X, and let x be a point on X. We show that the set S consisting of all nonnegative integers n such that f^n(x) is in Y is a union of at most…

We study a parametric family of piecewise rotations of the torus, in the limit in which the rotation number approaches the rational value 1/4. There is a region of positive measure where the discontinuity set becomes dense in the limit; we…

Hans J. Zassenhaus conjectured that for any unit $u$ of finite order in the integral group ring of a finite group $G$ there exists a unit $a$ in the rational group algebra of $G$ such that $a^{-1}\cdot u \cdot a=\pm g$ for some $g\in G$. We…

We obtain two sequences of rational numbers which converge to the Euler-Gompertz constant. Denote by <f(x)> the integral of f(x)e^{-x} from 0 to infinity. Recall that the Euler-Gompertz constant \delta is <ln(x+1)>. Main idea. Let P_n(x) be…

Define the map $\mathsf{T}$ on the positive integers by $\mathsf{T}(m)=\frac{m}{2}$ if $m$ is even and by $\mathsf{T}(m)=\frac{3m+1}{2}$ if $m$ is odd. Results of Terras and Everett imply that, given any $\epsilon>0$, almost all…

We demonstrate the impact of a generic zero-free region and zero-density estimate on the error term in the prime number theorem. Consequently, we are able to improve upon previous work of Pintz and provide an essentially optimal error term…

The $3x+1$ Conjecture asserts that the $T$-orbit of every positive integer $x$ contains $1$, where $T$ maps $x$ to $x/2$ for $x$ even and to $(3x+1)/2$ for $x$ odd. Several authors have studied the analogous map, $T_q$, which maps $x\in…

Let $M$ be a compact closed manifold of variable negative curvature. Fix an element $\operatorname{id} \neq \gamma$ in the fundamental group $\Gamma$ of $M$, and denote the set of elements in $\Gamma$ that are conjugate to $\gamma$ by…

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…

The Golomb-Welch conjecture states that there are no perfect $e$-error-correcting Lee codes in $\mathbb{Z}^n$ ($PL(n,e)$-codes) whenever $n\geq 3$ and $e\geq 2$. A special case of this conjecture is when $e=2$. In a recent paper of A.…

A longstanding conjecture of Erd\H{o}s and Simonovits states that for every rational $r$ between $1$ and $2$ there is a graph $H$ such that the largest number of edges in an $H$-free graph on $n$ vertices is $\Theta(n^r)$. Answering a…

An attempt to come closer to a resolution of the Collatz conjecture is presented. The central idea is the formation of a tree consisting of positive odd numbers with number 1 as root. Functions for generating the tree from the root are…