On Sums of "Small" Cantor Sets — Draconitic Research

Abstract

We provide several results showing that sums of ``small'' Cantor sets can turn into an interval or a finite union of intervals. Specifically, we show that 1) There exists a Cantor set C⊂ [0,1] of zero logarithmic capacity such that C+C=[0,2]. 2) For any n≥ 2 there exists a Cantor set C⊂ [0,1] of zero logarithmic capacity such that a Minkowski sum of n copies of C is an interval [0,n], while a sum of any smaller number of copies of C is still a Cantor set. 3) For any n≥ 2 there exists a collection of Cantor sets C₁, C₂, …, C_n⊂ [0,1] of zero lower box-counting dimension such that a Minkowski sum of any n−1 sets from this collection is a Cantor set of zero lower box-counting dimension, while the sum of all n sets is an interval [0,n]. 4) There is an uncountable family of Cantor sets of zero lower box-counting dimension such that the sum of any two different sets from this family is a finite union of intervals. 5) For each of the properties above, there are examples of Cantor sets with the same property having zero Hausdorff h-measure for an arbitrary given gauge function h.

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BibTeX RIS

BibTeX

@misc{gorodetski2026sums,
  title = {{On Sums of "Small" Cantor Sets}},
  author = {Anton Gorodetski and Alexandro Luna and William Wood},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.17005},
  eprint = {2610.17005},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.17005},
}

RIS

TY  - UNPB
TI  - On Sums of "Small" Cantor Sets
AU  - Anton Gorodetski
AU  - Alexandro Luna
AU  - William Wood
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.17005
UR  - https://arxiv.org/abs/2610.17005
AB  - We provide several results showing that sums of ``small'' Cantor sets can turn into an interval or a finite union of intervals. Specifically, we show that 1) There exists a Cantor set C⊂ [0,1] of zero logarithmic capacity such that C+C=[0,2]. 2) For any n≥ 2 there exists a Cantor set C⊂ [0,1] of zero logarithmic capacity such that a Minkowski sum of n copies of C is an interval [0,n], while a sum of any smaller number of copies of C is still a Cantor set. 3) For any n≥ 2 there exists a collection of Cantor sets C_1, C_2, …, C_n⊂ [0,1] of zero lower box-counting dimension such that a Minkowski sum of any n−1 sets from this collection is a Cantor set of zero lower box-counting dimension, while the sum of all n sets is an interval [0,n]. 4) There is an uncountable family of Cantor sets of zero lower box-counting dimension such that the sum of any two different sets from this family is a finite union of intervals. 5) For each of the properties above, there are examples of Cantor sets with the same property having zero Hausdorff h-measure for an arbitrary given gauge function h.
ER  -

Abstract

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