PreprintCC BY 4.0arXiv
Attainable forms of intermediate dimensions of complementary sets
Abstract
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↦dim̅ θE arise as the upper intermediate dimensions of sets E∈𝒞a. 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 dim̅BCa 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.
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BibTeX
@misc{angelini2026attainable,
title = {{Attainable forms of intermediate dimensions of complementary sets}},
author = {Nicolas Angelini},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35834},
eprint = {2610.35834},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35834},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Attainable forms of intermediate dimensions of complementary sets
AU - Nicolas Angelini
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35834
UR - https://arxiv.org/abs/2610.35834
AB - Given a positive, non-increasing sequence a with sum $1$, we consider the family 𝒞_a of closed subsets of [0,1] whose complementary open intervals have lengths given by the terms of a. We study which functions θ↦dim̅_ θE arise as the upper intermediate dimensions of sets E∈𝒞_a. 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 dim̅_BC_a at θ=1 and are bounded above by the upper intermediate dimensions of the decreasing Cantor set C_a. We also show that the lower bound given by the countable set D_a is rigid: if it is attained at one value of θ, then it is attained at all larger values.
ER -