Arbitrarily long strings of consecutive primes in special sets

Let $F(x)$ be a function of the form $ \sum_{i=1}^r d_i x^{\rho_i}$, where $d_1,\ldots ,d_r\in \mathbb {R}$, $0 \leqslant \rho_1 \lt \cdots \lt \rho_r$, $\rho _r \notin \mathbb Z$, $\rho _i \in \mathbb R$ for $ 1 \leqslant i \leqslant r$ and $d_r\ne 0$. We prove that the 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.