Roth's Theorem in the Piatetski-Shapiro primes

Let $\mathbf{P}$ denote the set of prime numbers and, for an appropriate function $h$, define a set $\mathbf{P}_{h}=\{p\in\mathbf{P}: \exists_{n\in\mathbb{N}}\ p=\lfloor h(n)\rfloor\}$. The aim of this paper is to show that every subset of $\mathbf{P}_{h}$ having positive relative upper density contains a nontrivial three-term arithmetic progression. In particular the set of Piatetski--Shapiro primes of fixed type $71/72<\gamma<1$, i.e. $\{p\in\mathbf{P}: \exists_{n\in\mathbb{N}}\ p=\lfloor n^{1/\gamma}\rfloor\}$ has this feature. We show this by proving the counterpart of Bourgain--Green's restriction theorem for the set $\mathbf{P}_{h}$.Comment: Accepted for publication in Revista Matematica Iberoamerican

On fixed points of a generalized multidimensional affine recursion

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 matrix $A$, we consider the following generalized multidimensional affine equation R\stackrel{\mathcal{D}}{=}\sum_{i=1}^N A_iR_i+B, where $N\ge2$ is a fixed natural number, $A_1,...,A_N$ are independent copies of $A$, $B\in\mathbb{R}^d$ is a random vector with positive entries, and $R_1,...,R_N$ are independent copies of $R\in\mathbb{R}^d$, which have also positive entries. Moreover, all of them are mutually independent and $\stackrel{\mathcal{D}}{=}$ stands for the equality in distribution. We will show with the aid of spectral theory developed by Guivarc'h and Le Page and Kesten's renewal theorem, that under appropriate conditions, there exists $\chi>0$ such that $\P(\{>t\})\asymp t^{-\chi},$ as t\to\8, for every unit vector $u\in\mathbb{S}^{d-1}$ with positive entries.Comment: 30 pages, no figure

Asymptotics of stationary solutions of multivariate stochastic recursions with heavy tailed inputs and related limit theorems

Let $\Phi_n$ be an i.i.d. sequence of Lipschitz mappings of $\R^d$. We study the Markov chain $\{X_n^x\}_{n=0}^\infty$ on $\R^d$ defined by the recursion $X_n^x = \Phi_n(X^x_{n-1})$, $n\in\N$, $X_0^x=x\in\R^d$. We assume that $\Phi_n(x)=\Phi(A_n x,B_n(x))$ for a fixed continuous function $\Phi:\R^d\times \R^d\to\R^d$, commuting with dilations and i.i.d random pairs $(A_n,B_n)$, where $A_n\in {End}(\R^d)$ and $B_n$ is a continuous mapping of $\R^d$. Moreover, $B_n$ is $\alpha$-regularly varying and $A_n$ has a faster decay at infinity than $B_n$. We prove that the stationary measure $\nu$ of the Markov chain $\{X_n^x\}$ is $\alpha$-regularly varying. Using this result we show that, if $\alpha<2$, the partial sums $S_n^x=\sum_{k=1}^n X_k^x$, appropriately normalized, converge to an $\alpha$-stable random variable. In particular, we obtain new results concerning the random coefficient autoregressive process $X_n = A_n X_{n-1}+B_n$.Comment: 23 pages, 0 figures. Accepted for publication in Stochastic Processes and their Application