Binary refined Diophantine exponent spectrum

Determine whether the spectrum of the refined Diophantine exponent over a binary alphabet is the entire interval [1,∞].

Background

The paper proves that over a ternary alphabet the spectrum of the refined Diophantine exponent is [1,∞]. It also establishes density results for words with prescribed refined Diophantine exponent when the alphabet has at least three letters. Whether the same full spectrum can be achieved over a binary alphabet is left unresolved.