Related papers: Multiple recurrence and convergence for certain av…
The prime numbers have been a source of fascination for millenia and continue to surprise us. Motivated by the hyperuniformity concept, which has attracted recent attention in physics and materials science, we show that the prime numbers in…
The arithmetic average of the first $n$ primes, $\bar p_n = {1\over n} \sum_{i=1}^n p_i$, exhibits very many interesting and subtle properties. Since the transformation from $p_n \to \bar p_n$ is extremely easy to invert, $p_n = n\bar p_n -…
Given an affine variety $X$, a morphism $\phi:X\to X$, a point $\alpha\in X$, and a Zariski closed subset $V$ of $X$, we show that the forward $\phi$-orbit of $\alpha$ meets $V$ in at most finitely many infinite arithmetic progressions, and…
We generalize Petridis's new proof of Pl\"unnecke's graph inequality to graphs whose vertex set is a measure space. Consequently, this gives new Pl\"unnecke inequalities for measure preserving actions which enable us to deduce, via a…
We give a short proof of the fact that every set of natural numbers with positive upper Banach density contains the sum of two infinite sets. The approach simplifies earlier existing proofs.
The Collatz Conjecture's connection to dynamical systems opens it to a variety of techniques aimed at recurrence and density results. First, we turn to density results and strengthen the result of Terras through finding a strict rate of…
We introduce a family of maps generating continued fractions where the digit $1$ in the numerator is replaced cyclically by some given non-negative integers $(N_1,\ldots,N_m)$. We prove the convergence of the given algorithm, and study the…
Let $\delta > 1/2$. We prove that if $A$ is a subset of the primes such that the relative density of $A$ in every reduced residue class is at least $\delta$, then almost all even integers can be written as the sum of two primes in $A$. The…
The culmination of the papers (arXiv:0905.0518, arXiv:0910.0909) was a proof of the norm convergence in $L^2(\mu)$ of the quadratic nonconventional ergodic averages \frac{1}{N}\sum_{n=1}^N(f_1\circ T_1^{n^2})(f_2\circ…
Motivated by a recent investigation of Costakis et al. on the notion of recurrence in linear dynamics, we study various stronger forms of recurrence for linear operators, in particular that of frequent recurrence. We study, among other…
Prime number multiplet classifications and patterns are extended to negative integers. The extension from prime numbers to single prime powers is also studied. Prime number septets at equal distance are given. It is also shown that each…
In this paper, we deal with a notion of Banach space-valued mappings defined on a set consisting of finite graphs with uniformly bounded vertex degree. These functions will be endowed with certain boundedness and additivity criteria. We…
For any positive integer $n$ along with parameters $\alpha$ and $\nu$, we define and investigate $\alpha$-shifted, $\nu$-offset, floor sequences of length $n$. We find exact and asymptotic formulas for the number of integers in such a…
In this paper we show that every set $A \subset \mathbb{N}$ with positive density contains $B+C$ for some pair $B,C$ of infinite subsets of $\mathbb{N}$, settling a conjecture of Erd\H{o}s. The proof features two different decompositions of…
We study logical limit laws for preferential attachment random graphs. In this random graph model, vertices and edges are introduced recursively: at time $1$, we start with vertices $0,1$ and $m$ edges between them. At step $n+1$ the vertex…
We study nonparametric Bayesian models for reversible multi-dimensional diffusions with periodic drift. For continuous observation paths, reversibility is exploited to prove a general posterior contraction rate theorem for the drift…
There are several extensions of the classical Banach Fixed Point Theorem in technical literature. A branch of generalizations replaces usual contractivity by weaker but still effective assumptions. Our note follows this stream, presenting…
The martingale posterior framework is a generalization of Bayesian inference where one elicits a sequence of one-step ahead predictive densities instead of the likelihood and prior. Posterior sampling then involves the imputation of unseen…
Abstract upper densities are monotone and subadditive functions from the power set of positive integers to the unit real interval that generalize the upper densities used in number theory, including the upper asymptotic density, the upper…
We study mean convergence results for weighted multiple ergodic averages defined by commuting transformations with iterates given by integer polynomials in several variables. Roughly speaking, we prove that a bounded sequence is a good…