Arbitrarily long strings of consecutive primes in special sets

Let F(x) be a function of the form ∑ri=1dixρi, where d1,…,dr∈R, 0≤ρ1<⋯<ρr, ρr∉Z, ρi∈R for 1≤i≤r and dr≠0. We prove that the sets of the form {n∈N:{F(n)}∈U} for any non-empty open set U⊂[0,1) contain arbitrarily long strings of consecutive primes.