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

Ziran Liu

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arXiv · 2609.00399

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

Ziran Liu

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Abstract

Let $X_1,X_2,\ldots$ be i.i.d. finitely supported random variables in a torsion-free abelian group, and write $S_k=X_1+\cdots+X_k$, and $H(S_k)$ is the Shannon entropy $S_k$, for all $k \ge 1$. We prove that, for every fixed $n\geq1$, \[ H(S_{n+1})-H(S_n) \geq \frac12\log\frac{n+1}{n} -o_{H(X_1)\to\infty}(1), \] uniformly over the ambient group and the input law. This proves a conjecture of Tao [29] in 2010.

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Ziran Liu. 2026-08-31. Shannon's problem on the monotonicity of entropy and a Conjecture of Tao. https://arxiv.org/abs/2609.00399

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