Same Gaps, Different Sets: A Survey on Complimentary sets and their Dimensions

Abstract

Let a=\{a_n\} be a non-increasing sequence of positive real numbers adding up to $1$. Cutting out of the unit interval disjoint open gaps of lengths $a_1,a_2,\dots$, one for each term, leaves a compact set of Lebesgue measure zero, and since the gaps can be placed in infinitely many ways, there are infinitely many such sets. We call them the \emph{complementary sets} of $a$ and write $\mathfrak{C}_a$ for the family of all of them. Every set in $\mathfrak{C}_a$ is compact and totally disconnected, and all of them share the same gap lengths; what distinguishes one from another is only the arrangement of the gaps. Among them, the perfect sets are exactly the Cantor sets with those gaps, and they form a distinguished subclass. The central question of this survey is what such a rearrangement can change and what it cannot. The answer depends sharply on the notion of size: the box dimensions are constant on $\mathfrak{C}_a$, while the Hausdorff dimension attains every value in a closed interval whose endpoints are realised by two distinguished members, the Cantor set $C_a$ built by placing the gaps in a Cantor-like pattern, and the countable decreasing rearrangement $D_a$. We survey how this dichotomy persists, and how the roles of $C_a$ and $D_a$ are exchanged, across the intermediate dimensions of Falconer, Fraser and Kempton, the Assouad-type and $Φ$-dimensions, and the Hausdorff and packing $h$-measures associated with a general dimension function; we also describe the almost sure behaviour of a randomly rearranged complementary set, where a sharp threshold in the depth of the scales examined separates two incompatible regimes. A final part restricts attention to the Cantor subclass and turns to the dynamical description of the sets C_a.