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PreprintarXiv licencearXiv

Shannon's problem on the monotonicity of entropy and a Conjecture of Tao

  • Ziran Liu
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Abstract

Let X1,X2,… be i.i.d. finitely supported random variables in a torsion-free abelian group, and write Sk=X1+⋯+Xk, and H(Sk) is the Shannon entropy Sk, for all k ≥ 1. We prove that, for every fixed n≥1, H(Sn+1)−H(Sn) ≥ 1/2log(n+1)/n −oH(X1)→∞(1), uniformly over the ambient group and the input law. This proves a conjecture of Tao [29] in 2010.

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Cite

BibTeXRIS
BibTeX
@misc{liu2026shannons,
  title = {{Shannon's problem on the monotonicity of entropy and a Conjecture of Tao}},
  author = {Ziran Liu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.00399},
  eprint = {2609.00399},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.00399},
}
RIS
TY  - UNPB
TI  - Shannon's problem on the monotonicity of entropy and a Conjecture of Tao
AU  - Ziran Liu
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.00399
UR  - https://arxiv.org/abs/2609.00399
AB  - Let X_1,X_2,… be i.i.d. finitely supported random variables in a torsion-free abelian group, and write S_k=X_1+⋯+X_k, and H(S_k) is the Shannon entropy S_k, for all k ≥ 1. We prove that, for every fixed n≥1, H(S_n+1)−H(S_n) ≥ 1/2log(n+1)/n −o_H(X_1)→∞(1), uniformly over the ambient group and the input law. This proves a conjecture of Tao [29] in 2010.
ER  -