Arbitrarily long strings of consecutive primes in special sets

Abstract

Let $F(x)$ be a function of the form ${\sum }_{i=1}^r{d}_i{x}^{{\rho }_i}$ where $d_1,\dots ,d_r\in \mathbb{R}$, $0\le {\rho }_1<\dots <{\rho }_r,$ ${\rho }_r\notin \mathbb{Z},{\rho }_i\in \mathbb{R}$ for $1\le i\le r$ and $d_r\ne 0$. We prove that sets of the form $\{n\in \mathbb{N}:\{F(n)\}\in U\}$ for any non-empty open set $U\subset [0,1)$ contain arbitrarily long strings of consecutive primes.

arXiv commentv2: 35 pages, incorporated changes suggested by referees