Related papers: On the concentration of certain additive functions

Let $\{p_j(n)\}_{j=1}^{\omega(n)}$ denote the increasing sequence of distinct prime factors of an integer $n$. For $z\geqslant 0$, let $G(n;z)$ denote the number of those indexes $j$ such that $p_{j+1}(n)>p_j(n)^{\exp z}$. We show uniform…

In this paper, we study a class of Cantor-integers $\{C_n\}_{n\geq 1}$ with the base conversion function $f:\{0,\dots,m\}\to \{0,\dots,p\}$ being strictly increasing and satisfying $f(0)=0$ and $f(m)=p$. Firstly we provide an algorithm to…

The paper provides a simpler method for proving a delicate inequality that was used by Achlioptis and Naor to establish asymptotic concentration for chromatic numbers of Erdos-Renyi random graphs. The simplifications come from two new…

Let $u(x)$ be a subpolynomial function in a Hardy field. We establish necessary and sufficient conditions for the weighted uniform distribution of the sequences $(u(n))_{n\in\mathbb{N}}$ and $(u(p_n))_{n\in\mathbb{N}}$, where $p_n$ denotes…

Heavy particles in turbulent flows have been shown to accumulate in regions of high strain rate or low vorticity, a process otherwise known as preferential concentration. This can be observed in geophysical flows, and is inferred to occur…

Given a sequence $\mathscr{A}=\{a_0<a_1<a_2\ldots\}\subseteq \mathbb{N}$, let $r_{\mathscr{A},h}(n)$ denote the number of ways $n$ can be written as the sum of $h$ elements of $\mathscr{A}$. Fixing $h\geq 2$, we show that if $f$ is a…

We will study the asymptotic behavior of summation functions of a natural argument, including the asymptotic behavior of summation functions of a prime argument in the paper. A general formula is obtained for determining the asymptotic…

Consider a uniform expanders family G_n with a uniform bound on the degrees. It is shown that for any p and c>0, a random subgraph of G_n obtained by retaining each edge, randomly and independently, with probability p, will have at most one…

Let $a>1$. Denote by $l_a(p)$ the multiplicative order of $a$ modulo $p$. We look for an estimate of sum of $\frac{l_a(p)}{p-1}$ over primes $p\leq x$ on average. When we average over $a\leq N$, we observe a statistic of $C\mathrm{Li}(x)$.…

Let $s(n)$ denote the number of ones in the binary expansion of the nonnegative integer $n$. How does $s$ behave under addition of a constant $t$? In order to study the differences \[s(n+t)-s(n),\] for all $n\ge0$, we consider the…

Let $P$ be a subset of the primes of lower density strictly larger than $\frac12$. Then, every sufficiently large even integer is a sum of four primes from the set $P$. We establish similar results for $k$-summands, with $k\geq 4$, and for…

Let A be finite set equipped with a probability distribution P, and let M be a "mass" function on A. A characterization is given for the most efficient way in which A^n can be covered using spheres of a fixed radius. A covering is a subset…

Let $\{f_i:\mathbb{F}_p^i \to \{0,1\}\}$ be a sequence of functions, where $p$ is a fixed prime and $\mathbb{F}_p$ is the finite field of order $p$. The limit of the sequence can be syntactically defined using the notion of ultralimit.…

In this paper we further study the relationship between convexity and additive growth, building on the work of Schoen and Shkredov (\cite{SS}) to get some improvements to earlier results of Elekes, Nathanson and Ruzsa (\cite{ENR}). In…

We consider a finite or countable collection of one-dimensional Brownian particles whose dynamics at any point in time is determined by their rank in the entire particle system. Using Transportation Cost Inequalities for stochastic…

Using GCD sums, we show that the set of the primes has small common multiplicative energy with an arbitrary exponentially big integer set $S$ and, in particular, size of any arithmetic progression in $S$ having the beginning at zero, is at…

Let $A$ be an $n \times n$ matrix, $X$ be an $n \times p$ matrix and $Y = AX$. A challenging and important problem in data analysis, motivated by dictionary learning and other practical problems, is to recover both $A$ and $X$, given $Y$.…

We establish the exact-order estimates of uniform approximations by the Zygmund sums $Z^{s}_{n-1}$ of $2\pi$-periodic continuous functions $f$ from the classes $C^{\psi}_{\beta,p}$.These classes are defined by the convolutions of functions…

We investigate the aggregation kinetics of sedimenting particles theoretically and numerically, using the advection-diffusion equation. Agglomeration, caused by both transport mechanisms (diffusion and advection), is important for small…

A diffusion-limited aggregation process, in which clusters coalesce by means of 3-particle reaction, A+A+A->A, is investigated. In one dimension we give a heuristic argument that predicts logarithmic corrections to the mean-field asymptotic…