Binary words with a given Diophantine exponent

We prove that every real number ξ≥1 is the Diophantine exponent of some binary word ω. More precisely, we show that Dio(ω)=ξ for ω=10k110k210k3⋯, where kn=[ξn] for ξ≥2,kn=[νn] with for 1<ξ<2, and kn=n for ξ=1.