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PreprintCC BY-SA 4.0arXiv

Counting Residue-Class Survivor Sets: An Asymptotic Comparison with Prime-Admissible Sets

  • Mario Raso
  • Daniele Venturi
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Abstract

For n≥1, choose one residue class rkpmodk for every 2≤ k≤ n, and retain those m∈{2,…,n+1} for which mnot≡ rkpmodk whenever k<m. Let N(n) be the number of distinct survivor sets obtained in this way. We prove log 2≤ liminfn→∞(log N(n))/(n/log n) ≤ limsupn→∞(log N(n))/(n/log n) ≤2log 2. If Aadm(n) denotes the number of subsets of {2,…,n+1} that omit at least one residue class modulo every prime, then our main comparison is log N(n)=log Aadm(n)+o (n/(log n)). We also obtain an asymptotic formula for the number of distinct intersections of survivor sets with the primes, and an exact recurrence N(n+1)=N(n)+E(n), where E(n) counts the survivor sets that can be extended by the new point n+2.

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Cite

BibTeXRIS
BibTeX
@misc{raso2026counting,
  title = {{Counting Residue-Class Survivor Sets: An Asymptotic Comparison with Prime-Admissible Sets}},
  author = {Mario Raso and Daniele Venturi},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.08528},
  eprint = {2609.08528},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.08528},
  note = {CC BY-SA 4.0},
}
RIS
TY  - UNPB
TI  - Counting Residue-Class Survivor Sets: An Asymptotic Comparison with Prime-Admissible Sets
AU  - Mario Raso
AU  - Daniele Venturi
PY  - 2026
DA  - 2026/09/27
PB  - arXiv
DO  - 10.48550/arxiv.2609.08528
UR  - https://arxiv.org/abs/2609.08528
AB  - For n≥1, choose one residue class r_kpmodk for every 2≤ k≤ n, and retain those m∈{2,…,n+1} for which mnot≡ r_kpmodk whenever k<m. Let N(n) be the number of distinct survivor sets obtained in this way. We prove log 2≤ liminf_n→∞(log N(n))/(n/log n) ≤ limsup_n→∞(log N(n))/(n/log n) ≤2log 2. If A_adm(n) denotes the number of subsets of {2,…,n+1} that omit at least one residue class modulo every prime, then our main comparison is log N(n)=log A_adm(n)+o (n/(log n)). We also obtain an asymptotic formula for the number of distinct intersections of survivor sets with the primes, and an exact recurrence N(n+1)=N(n)+E(n), where E(n) counts the survivor sets that can be extended by the new point n+2.
ER  -