Random Survivor Sets with Moving Holes in Bedford-McMullen Carpets

@misc{kong2026random,
  title = {{Random Survivor Sets with Moving Holes in Bedford-McMullen Carpets}},
  author = {Derong Kong and Cai-Yun Ma and Károly Simon},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.31440},
  eprint = {2610.31440},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.31440},
}

TY  - UNPB
TI  - Random Survivor Sets with Moving Holes in Bedford-McMullen Carpets
AU  - Derong Kong
AU  - Cai-Yun Ma
AU  - Károly Simon
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.31440
UR  - https://arxiv.org/abs/2610.31440
AB  - Let 2≤ m < n be integers, and let K(𝒜) be the Bedford--McMullen carpet generated by {(x,y)↦((x+i)/m,(y+j)/n):(i,j)∈𝒜}, where 𝒜 is a subset of {0,1,…,m−1}×{0,1,…,n−1}. Write ℬ for the set of occupied columns, and assume that #ℬ≥2 and each column contains at least two symbols of 𝒜. At each vertex of the column tree ℬ^*, we independently choose one forbidden symbol from 𝒜 according to a prescribed distribution. Let ℙ denote the resulting product measure on 𝒜^ℬ^*. This gives a random survivor set K_ω(𝒜) with ω∈𝒜^ℬ^*. We show that, for ℙ-a.e. ω, dim_H K_ω(𝒜)=1/(log m)log(∫ ∑_b∈ℬℓ_b(ω)^(log m)/(log n) dℙ(ω)), where ℓ_b(ω) is the number of surviving symbols in column b at the root.
ER  -