bibkey: goh2026entropicenergy
authors: M. K. Goh
year: 2026
title: On an entropic analogue of additive energy
doi: 10.2140/ent.2026.5.243
url: https://arxiv.org/abs/2406.18798v4
claim: "Conjecture 15 (89) asks for an unconditional entropic Glibichuk–Konyagin lemma."
strata_touched:
  - D5/S3/Entropy/GohGlibichukKonyagin/Result
license: citation-only
triage: anchor

Entropic Glibichuk–Konyagin conjecture

Verified locator

https://arxiv.org/abs/2406.18798v4

M. K. Goh, On an entropic analogue of additive energy, Essential Number Theory 5 (2026), no. 2, 243–269. DOI 10.2140/ent.2026.5.243. The statement is in §6, Conjecture 15, equation (89). The paper uses binary logarithms and discrete random variables.

Source statement

Conjecture 15 (Entropic Glibichuk–Konyagin lemma). There is an integer $k\ge 3$ such that the following holds. Let $X$ be a random variable taking values in ${\mathbf{F}}_p$ for a prime $p$. Then for independent copies $X_1,\dots ,X_k,X_1^{\prime },\dots ,X_k^{\prime }$ of $X$, we have $$H\{X_1{X}_1^{\prime }+\cdots +X_k{X}_k^{\prime }\}\ge \min (2H\{X\},\log p)-1.(89)$$

In the original additive-combinatorial version of this statement, $k$ can be taken to be $3$, but any fixed integer is enough for our purposes.

No restriction to uniform laws or upper bound on the largest atom is stated. Conjecture 15 is used in the source’s derivation of Conjecture 14.