Well-approximable coefficients in monomial bracket words

Determine whether, for every irrational well-approximable number β, every integer ℓ≥2, and every nonconstant piecewise-constant coding function with partition boundaries on [0,1), the monomial bracket word defined by $a_n=c(\{\beta n^\ell\})$ has infinite Diophantine exponent.

Background

For the monomial bracket word $a_n=c(\{\beta n^\ell\})$, the paper proves that the ordinary Diophantine exponent is infinite under the stronger condition $\limsup q_{n+1}/q_n^\ell=\infty$, where $p_n/q_n$ are the continued-fraction convergents of β. The paper observes that this condition is stronger than mere well approximability and explains that the dependence of the shift $g(i+q_n)-g(i)$ on i makes a full dichotomy difficult. It therefore leaves open whether well approximability alone suffices.