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PreprintCC BY-SA 4.0arXiv

Dimension-free exponential entropy quotients over 𝔽2

  • Josiah Park
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Abstract

Let X be a random vector in 𝔽2m, and let X' be an independent copy. We prove that, for every ε>0, there is a subspace G with dim G≤ 2320(1+ε)/εH(X) such that H(qGX)≤(1+ε)(H(qGX+qGX')−H(qGX)). In particular, the quotient entropy is at most (1+ε)(H(X+X')−H(X)). The coefficient is independent of the ambient dimension, and its exponential order in 1/ε is necessary. The structural result is a logarithmic rigidity estimate for finite weighted families of laws with averaged subspace entropy sparsity. A common variational quotient controls both a family and its convolution family without decreasing the original family's normalized mean entropy defect. Posterior families preserve averaged sparsity, and a fresh-index argument combines a local-to-global estimate with the entropic Marton theorem to increase the defect ratio at a fixed multiplicative sparsity cost. We also obtain two-input almost-additivity and concentration of moderate-doubling sets on subspace cosets at the same exponential scale. The constants are not optimized.

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Cite

BibTeXRIS
BibTeX
@misc{park2026dimensionfree,
  title = {{Dimension-free exponential entropy quotients over $\mathbbF_2$}},
  author = {Josiah Park},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.38525},
  eprint = {2610.38525},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.38525},
  note = {CC BY-SA 4.0},
}
RIS
TY  - UNPB
TI  - Dimension-free exponential entropy quotients over 𝔽_2
AU  - Josiah Park
PY  - 2026
DA  - 2026/10/28
PB  - arXiv
DO  - 10.48550/arxiv.2610.38525
UR  - https://arxiv.org/abs/2610.38525
AB  - Let X be a random vector in 𝔽_2^m, and let X' be an independent copy. We prove that, for every ε>0, there is a subspace G with dim G≤ 2^320(1+ε)/εH(X) such that H(q_GX)≤(1+ε)(H(q_GX+q_GX')−H(q_GX)). In particular, the quotient entropy is at most (1+ε)(H(X+X')−H(X)). The coefficient is independent of the ambient dimension, and its exponential order in 1/ε is necessary. The structural result is a logarithmic rigidity estimate for finite weighted families of laws with averaged subspace entropy sparsity. A common variational quotient controls both a family and its convolution family without decreasing the original family's normalized mean entropy defect. Posterior families preserve averaged sparsity, and a fresh-index argument combines a local-to-global estimate with the entropic Marton theorem to increase the defect ratio at a fixed multiplicative sparsity cost. We also obtain two-input almost-additivity and concentration of moderate-doubling sets on subspace cosets at the same exponential scale. The constants are not optimized.
ER  -