Attainable forms of intermediate dimensions of complementary sets

Given a positive, non-increasing sequence a with sum $1$, we consider the family Cₐ of closed subsets of $[0,1]$ whose complementary open intervals have lengths given by the terms of a. We study which functions θ↦̄\,θE arise as the upper intermediate dimensions of sets Eₐ. Under a natural condition on the decay of a, we show that these are exactly the functions in the class introduced by Banaji and Rutar that take the value ̄BCₐ at $θ=1$ and are bounded above by the upper intermediate dimensions of the decreasing Cantor set Cₐ. We also show that the lower bound given by the countable set Dₐ is rigid: if it is attained at one value of $θ$, then it is attained at all larger values.