Related papers: Positive-definite Functions, Exponential Sums and …

Let $f$ be a real-valued $1$-bounded multiplicative function. Suppose that the mean-value of $f^{2}$ exists, and $$\int_{0}^{1} \Big | \sum_{n \leq N} f(n)e^{2\pi i n \alpha} \Big | d \alpha\leq N^{o(1)}$$ as $N \rightarrow \infty$, then…

Let $F_1=1,F_2=1,\ldots$ be the Fibonacci sequence. Motivated by the identity $\displaystyle\sum_{k=0}^{\infty}\frac{1}{F_{2^k}}=\frac{7-\sqrt{5}}{2}$, Erd\"os and Graham asked whether $\displaystyle\sum_{k=1}^{\infty}\frac{1}{F_{n_k}}$ is…

We consider a system $\displaystyle \frac{dx}{dt}=r_1(t) G_1(x) \left[ \int_{h_1(t)}^t f_1(y(s))~d_s R_1 (t,s) - x(t) \right], \frac{dy}{dt}=r_2(t) G_2(y) \left[ \int_{h_2(t)}^t f_2(x(s))~d_s R_2 (t,s) - y(t)\right]$ with increasing…

In this paper we introduce several extremal sequences of points on locally compact metric spaces and study their asymptotic properties. These sequences are defined through a greedy algorithm by minimizing a certain energy functional whose…

Let $n,d$, and $k$ be positive integers where $n$ and $d$ are coprime. Our two main results are Theorem 1. There is a partition of the infinite interval $[kd,\infty)$ of positive integers into a family of finite sets $X$ for which the sum…

The origin of deterministic diffusion is a matter of discussion. We study the asymptotic distributions of the sums $y_n(x)=\sum_{k=0}^{n-1}\psi (x+k\alpha)$, where $\psi$ is a periodic function of bounded variation and $\alpha$ an…

Let $G=(G_j)_{j\ge 0}$ be a strictly increasing linear recurrent sequence of integers with $G_0=1$ having characteristic polynomial $X^{d}-a_1X^{d-1}-\cdots-a_{d-1}X-a_d$. It is well known that each positive integer $\nu$ can be uniquely…

We investigate the density properties of generalized divisor functions $\displaystyle f_s(n)=\frac{\sum_{d|n}d^s}{n^s}$ and extend the analysis from the already-proven density of $s=1$ to $s\geq0$. We demonstrate that for every $s>0$, $f_s$…

We consider the amusing sequence of functions $f_n: \mathbb{R} \rightarrow \mathbb{R}$ given by $$ f_n(x) = \sum_{k=1}^{n}{\frac{|\sin{(k \pi x)}|}{k}}.$$ Every rational point is eventually the location of a strict local minimum of $f_n$:…

The aim of this paper is to develop greedy algorithms which generate uniformly distributed sequences in the $d$-dimensional unit cube $[0,1]^d$. The figures of merit are three different variants of $L_2$ discrepancy. Theoretical results…

Let E_n={x_i=1, x_i+x_j=x_k, x_i*x_j=x_k: i,j,k \in {1,...,n}}. We prove: (1) there is an algorithm that for every computable function f:N-->N returns a positive integer m(f), for which a second algorithm accepts on the input f and any…

Given a periodic function $f$, we study the almost everywhere and norm convergence of series $\sum_{k=1}^\infty c_k f(kx)$. As the classical theory shows, the behavior of such series is determined by a combination of analytic and number…

Let $f(n)$ be a Steinhaus random multiplicative function. Let $A\subset [1, N]$ be a finite set of integers. We show that \[\frac{1}{\sqrt{|A|}} \sum_{n\in A} f(n) \xrightarrow[]{d} \mathcal{CN}(0,1)\] forces that $|A|=o(N)$. We prove that…

We deduce an asymptotic formula with error term for the sum $\sum_{n_1,\ldots,n_k \le x} f([n_1,\ldots, n_k])$, where $[n_1,\ldots, n_k]$ stands for the least common multiple of the positive integers $n_1,\ldots, n_k$ ($k\ge 2$) and $f$…

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…

The article presents mathematical generalization of results which originated as solutions of practical problems, in particular, the modeling of transitional processes in electrical circuits and problems of resource allocation. However, the…

Let $\mathcal{G}$ be the greedy algorithm that, for each $\theta\in (0,1]$, produces an infinite sequence of positive integers $(a_n)_{n=1}^\infty$ satisfying $\sum_{n=1}^\infty 1/a_n = \theta$. For natural numbers $p < q$, let…

Determining functionals are tools to describe the finite dimensional long-term dynamics of infinite dimensional dynamical systems. There also exist several applications to infinite dimensional {\em random} dynamical systems. In these…

We prove that if $f(n)$ is a Steinhaus or Rademacher random multiplicative function, there almost surely exist arbitrarily large values of $x$ for which $|\sum_{n \leq x} f(n)| \geq \sqrt{x} (\log\log x)^{1/4+o(1)}$. This is the first such…

Let $(r_k)$ be an increasing sequence and $(w_k)$ a positive sequence. We study the following question: is it true that for every sequence $(a_k)$ satisfying $\sum_{k=0}^\infty |a_k|^2 w_k^2 < \infty$ there exists a function $f\in…