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Equal subset sums and close divisors

  • Jianfeng Hou
  • Hongbin Zhao
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Abstract

For k≥2, let αk be the supremum of the exponents a for which almost every integer n has k distinct divisors in a multiplicative interval of relative length (log n)−a. Select each positive integer i independently with probability 1/i, forming a random set A, and let βk be the supremum of the c<1 for which, with probability tending to one as D→∞, the set A∩[Dc,D] has k distinct subsets with the same sum. We prove that αk=βk/(1−βk), resolving a conjecture of Ford, Green and Koukoulopoulos [Invent. Math. 232 (2023), 1027--1160]. We also prove that their weak and strict entropy thresholds coincide. The proof combines flag refinement and entropy concavity with an upper bound for approximate subset sums that is uniform in arbitrary translations. A model with independent geometric prime exponents then transfers this bound to divisors.

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Cite

BibTeXRIS
BibTeX
@misc{hou2026equal,
  title = {{Equal subset sums and close divisors}},
  author = {Jianfeng Hou and Hongbin Zhao},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.26014},
  eprint = {2609.26014},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.26014},
}
RIS
TY  - UNPB
TI  - Equal subset sums and close divisors
AU  - Jianfeng Hou
AU  - Hongbin Zhao
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.26014
UR  - https://arxiv.org/abs/2609.26014
AB  - For k≥2, let α_k be the supremum of the exponents a for which almost every integer n has k distinct divisors in a multiplicative interval of relative length (log n)^−a. Select each positive integer i independently with probability 1/i, forming a random set A, and let β_k be the supremum of the c<1 for which, with probability tending to one as D→∞, the set A∩[D^c,D] has k distinct subsets with the same sum. We prove that α_k=β_k/(1−β_k), resolving a conjecture of Ford, Green and Koukoulopoulos [Invent. Math. 232 (2023), 1027--1160]. We also prove that their weak and strict entropy thresholds coincide. The proof combines flag refinement and entropy concavity with an upper bound for approximate subset sums that is uniform in arbitrary translations. A model with independent geometric prime exponents then transfers this bound to divisors.
ER  -