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 = 1 0 k 1 1 0 k 2 1 0 k 3 ⋯ , where k n = [ ξ n ] for ξ ≥ 2 , k n = [ ν n ] with ν = ( − ξ + 1 + 6 ξ − 3 ξ 2 + 1 ) / ( 4 − 2 ξ ) for 1 < ξ < 2 , and k n = n for ξ = 1 .