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

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