Intermediate dimensions of complementary sets

Given a positive, non-increasing sequence a $a$ a with finite sum equal to 1 $1$ 1 , we consider the family of all closed subsets of left bracket 0 comma 1 right bracket $[0,1]$ [ 0 , 1 ] whose complementary open intervals have lengths given by a rearrangement of the sequence a $a$ a . We study the full range of possible theta $\theta$ θ -intermediate dimensions of these sets and, under suitable assumptions on the sequence, we show that this range forms a closed interval, whose endpoints we compute explicitly. This paper fills a gap in the literature concerning the dimensional properties of complementary sets.