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 a1,a2,…, 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 Ca for the family of all of them. Every set in Ca 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 Ca, while the Hausdorff dimension attains every value in a closed interval whose endpoints are realised by two distinguished members, the Cantor set Ca built by placing the gaps in a Cantor-like pattern, and the countable decreasing rearrangement Da. We survey how this dichotomy persists, and how the roles of Ca and Da 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.
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BibTeX
@misc{angelini2026same,
title = {{Same Gaps, Different Sets: A Survey on Complimentary sets and their Dimensions}},
author = {Nicolás Angelini and Carlos Cabrelli and Ursula Molter},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28960},
eprint = {2610.28960},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28960},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Same Gaps, Different Sets: A Survey on Complimentary sets and their Dimensions
AU - Nicolás Angelini
AU - Carlos Cabrelli
AU - Ursula Molter
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28960
UR - https://arxiv.org/abs/2610.28960
AB - 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,…, 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 C_a for the family of all of them. Every set in 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 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.
ER -