Related papers: Partial sums of biased random multiplicative funct…

In this note we compare two measures of the complexity of a class $\mathcal F$ of Boolean functions studied in (unconditional) pseudorandomness: $\mathcal F$'s ability to distinguish between biased and uniform coins (the coin problem), and…

Assume that the Riemann hypothesis holds for Dedekind zeta functions. Under this assumption, we prove that a degree $d$ polynomial with random multiplicative $\pm1$ coefficients is irreducible in $\mathbb{Z}[x]$ with probability…

A remarkable result of Bergelson, Tao and Ziegler implies that if $c>0$, $k$ is a positive integer, $p\geq k$ is a prime, $n$ is sufficiently large, and $f:\mathbb F_p^n\to\mathbb C$ is a function with $\|f\|_\infty\leq 1$ and…

A function $f: \mathbb{R}^d \rightarrow \mathbb{R}$ is referred to as a Sparse Additive Model (SPAM), if it is of the form $f(\mathbf{x}) = \sum_{l \in \mathcal{S}}\phi_{l}(x_l)$, where $\mathcal{S} \subset [d]$, $|\mathcal{S}| \ll d$.…

The main question of this paper is the following: how much cancellation can the partial sums restricted to the $k$-free integers up to $x$ of a $\pm 1$ multiplicative function $f$ be in terms of $x$? Building upon the recent paper by Q.…

Let $n$ be a positive integer and let $A$ be nonempty finite set of positive integers. We say that $A$ is relatively prime if $\gcd(A) =1$ and that $A$ is relatively prime to $n$ if $\gcd(A,n)=1$. In this work we count the number of…

We show that for a Steinhaus random multiplicative function $f:\mathbb{N}\to\mathbb{D}$ and any polynomial $P(x)\in\mathbb{Z}[x]$ of $\text{deg}\ P\ge 2$ which is not of the form $w(x+c)^{d}$ for some $w\in \mathbb{Z}$, $c\in \mathbb{Q}$,…

Let $E$ be an arbitrary subset of the unit circle $T$ and let $f$ be a function defined on $E$. When there exist polynomials $P_n$ which are uniformly bounded by a number $M > 0$ on $T$ and converge (pointwise) to $f$ at each point of $E$?…

We prove a functional central limit theorem for partial sums of symmetric stationary long range dependent heavy tailed infinitely divisible processes with a certain type of negative dependence. Previously only positive dependence could be…

Let $(X_i,i\geq 1)$ be a sequence of i.i.d. random variables with values in $[0,1]$, and $f$ be a function such that $`E(f(X_1)^2)<+\infty$. We show a functional central limit theorem for the process $t\mapsto \sum_{i=1}^n f(X_i)1_{X_i\leq…

In this paper, the averaging principle is studied for a class of multiscale stochastic partial differential equations driven by $\alpha$-stable process, where $\alpha\in(1,2)$. Using the technique of Poisson equation, the orders of strong…

We establish pointwise convergence for nonconventional ergodic averages taken along $\lfloor p^c\rfloor$, where $p$ is a prime number and $c\in(1,4/3)$ on $L^r$, $r\in(1,\infty)$. In fact, we consider averages along more general sequences…

Let S be a subset of the unit disk, and let F(s) denote the class of completely multiplicative functions f such that f(p) is in S for all primes p. The authors' main concern is which numbers arise as mean-values of functions in F(s). More…

In this work, we consider the proportion of smooth (free of large prime factors) values of a binary form $F(X_1,X_2)\in\Z[X_1,X_2]$. In a particular case, we give an asymptotic equivalent for this proportion which depends on $F$. This is…

In our previous paper \cite{co1} we have shown that the theory of circulant matrices allows to recover the result that there exists $p+1$ Mutually Unbiased Bases in dimension $p$, $p$ being an arbitrary prime number. Two orthonormal bases…

Let $p \in (0, \infty)$ be a constant and let $\{\xi_n\} \subset L^p(\Omega, {\mathcal F}, \P)$ be a sequence of random variables. For any integers $m, n \ge 0$, denote $S_{m, n} = \sum_{k=m}^{m + n} \xi_k$. It is proved that, if there…

We study the limiting behavior of smooth linear statistics of the spectrum of random permutation matrices in the mesoscopic regime, when the permutation follows one of the Ewens measures on the symmetric group. If we apply a smooth enough…

We determine the behavior of multiplicative functions vanishing at a positive proportion of prime numbers in almost all short intervals. Furthermore we quantify "almost all" with uniform power-saving upper bounds, that is, we save a power…

We introduce the concept of partial Poisson structure on a manifold $M$ modelled on a convenient space. This is done by specifying a (weak) subbundle $T^{\prime}M$ of $T^{\ast}M$ and an antisymmetric morphism $P:T^{\prime}M\rightarrow TM$…

For $f$ a Rademacher or Steinhaus random multiplicative function, we prove that $$ \max_{\theta \in [0,1]} \frac{1}{\sqrt{N}} \Bigl| \sum_{n \leq N} f(n) \mathrm{e} (n \theta) \Bigr| \gg \sqrt{\log N} ,$$ asymptotically almost surely as $N…