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OP-7577Status unassessedCombinatorics

Entropy of sum of random variables.

Additive combinatorics and its applications · Historical workshop question

MSC 05 · Combinatorics

Editorial mapping of AIMPL source category: Combinatorics. http://aimpl.org/addcombapp/ Not an author-supplied MSC code.

Historical list · status unassessedAs of 2026-09-08Source & evidence ↓
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Problem statement

Show H(Sn)≥H(Sn−1)+log⁡2nn−1−o(1).H(S_n)\geq H(S_{n-1})+\log_2 \sqrt[]{\frac{n}{n-1}} - o(1).

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Background

Question context

Let GG be a torsion free abelian group. Suppose XX is a random variable on GG and SnS_n is sum of nn many i.i.di.i.d copies of XX. Let H(⋅)H(\cdot) be the discrete entropy function.

We have the following theorem by T. Tao:

H(S2)≥H(S1)+log⁡22−o(1)H(S_2)\geq H(S_1)+\log_2 \sqrt[]{2}-o(1) where the o(1)o(1) term goes to zero as H(X)H(X) grows to infinity.

Progress & source assessment

Historical workshop record. Current open/solved status has not been independently reassessed.

Status note in the source

This is not even known for n=3n=3.

References
  1. [1]Additive combinatorics and its applications link

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