Related papers: A Generalization of the GGR Conjecture

Zeckendorf's theorem states that every positive integer can be uniquely decomposed as a sum of nonconsecutive Fibonacci numbers. The distribution of the number of summands converges to a Gaussian, and the individual measures on gaps between…

Let $k$ be a real abelian number field and $p$ an odd prime not dividing $[k:\mathbb{Q}]$. For a natural number $d$, let $E_d$ denote the group of units of $k$ congruent to $1$ modulo $d$, $C_d$ the subgroup of $d$-circular units of $E_d$,…

In this paper we present some extensions of the Ailon-Rudnick Theorem, which says that if $f,g\in{\mathbb C}[T]$, then $\gcd(f^n-1,g^m-1)$ is bounded for all $n,m\ge 1$. More precisely, using a uniform bound for the number of torsion points…

Recently, Grynkiewicz et al. [{\it Israel J. Math.} {\bf 193} (2013), 359--398], using tools from additive combinatorics and group theory, proved necessary and sufficient conditions under which the linear congruence $a_1x_1+\cdots…

We prove that assuming the Generalized Riemann Hypothesis every even integer larger than $\exp(\exp(15.85))$ can be written as the sum of a prime number and a number that has at most two prime factors.

We prove that for every function $f:X\to Y$, where $X$ is a separable Banach space and $Y$ is a Banach space with RNP, there exists a set $A\in\tilde\mcA$ such that $f$ is Gateaux differentiable at all $x\in S(f)\setminus A$, where $S(f)$…

In the present paper we prove that under the assumption of the GRH (Generalized Riemann Hypothesis) each sufficiently large odd integer can be expressed as the sum of a prime and two isolated primes.

Let R denote the reals, and let h: R^n --> R be a continuous, piecewise-polynomial function. The Pierce-Birkhoff conjecture (1956) is that any such h is representable in the form sup_i inf_j f_{ij}, for some finite collection of polynomials…

An interesting characterization of the Fibonacci numbers is that, if we write them as $F_1 = 1$, $F_2 = 2$, $F_3 = 3$, $F_4 = 5, ...$, then every positive integer can be written uniquely as a sum of non-adjacent Fibonacci numbers. This is…

Let $f$ be a primitive positive definite integral binary quadratic form of discriminant $-D$ and let $\pi_f(x)$ be the number of primes up to $x$ which are represented by $f$. We prove several types of upper bounds for $\pi_f(x)$ within a…

We prove that there exist functions $f$ and $g$ such that for all positive integers $k$ and $d$, for every graph $G$ and every subset $A$ of the vertices of $G$, either $G$ contains $k$ $A$-paths such that vertices of different $A$-paths…

Let $G_1,\dots,G_m$ be independent copies of the standard gaussian random vector in $\mathbb{R}^d$. We show that there is an absolute constant $c$ such that for any $A \subset S^{d-1}$, with probability at least $1-2\exp(-c\Delta m)$, for…

Reed conjectured that for every $\varepsilon>0$ and every integer $\Delta$, there exists $g$ such that the fractional total chromatic number of every graph with maximum degree $\Delta$ and girth at least $g$ is at most…

Let $G$ be a multiplicative subsemigroup of the general linear group $\Gl(\mathbb{R}^d)$ which consists of matrices with positive entries such that every column and every row contains a strictly positive element. Given a $G$--valued random…

We show that Li's criterion equivalent to the Riemann hypothesis, viz. the statement that the sums k_n=Sum_rho(1-(1-1/rho)^n) over Riemann xi-function zeroes as well as all derivatives lambda_n=…

Around 2008, Schramm conjectured that the critical probabilities for Bernoulli bond percolation satisfy the following continuity property: If $(G_n)_{n\geq 1}$ is a sequence of transitive graphs converging locally to a transitive graph $G$…

We prove a sharp spectral generalization of the Cheeger--Gromoll splitting theorem. We show that if a complete non-compact Riemannian manifold $M$ of dimension $n\geq 2$ has at least two ends and \[ \lambda_1(-\gamma\Delta+\mathrm{Ric})\geq…

The generalised Gegenbauer functions of fractional degree (GGF-Fs), denoted by ${}^{r\!}G^{(\lambda)}_\nu(x)$ (right GGF-Fs) and ${}^{l}G^{(\lambda)}_\nu(x)$ (left GGF-Fs) with $x\in (-1,1),$ $\lambda>-1/2$ and real $\nu\ge 0,$ are special…

We investigate a fifty-year-old conjecture of Erd\H{o}s and Graham concerning whether the binomial coefficient ${n \choose k}$ with $1 \leq k \leq \frac{n}{2}$ must always have a divisor $\leq n$ that is ``close'' to $n$: that is, bigger…

For a graph $G$ and $p\in[0,1]$, we denote by $G_p$ the random sparsification of $G$ obtained by keeping each edge of $G$ independently, with probability $p$. We show that there exists a $C>0$ such that if $p\geq C(\log n)^{1/3}n^{-2/3}$…