Entropy lower bounds and sum-product phenomena

TL;DR

This paper establishes new entropy lower bounds for sums and products of random variables, extending sum-product phenomena to arbitrary fields and providing inequalities that relate additive and multiplicative entropies.

Contribution

It introduces novel entropy bounds over arbitrary fields, including a prime-field analogue of Tao's entropy power inequality and a sum-product inequality involving entropy and min-entropy.

Findings

Derived a prime-field analogue of Tao's entropy power inequality.

Proved a sum-product entropy inequality valid over any field.

Established a relation between additive and multiplicative entropy doubling.

Abstract

Various lower bounds are established for the entropy of sums, products and their combinations. First, we derive a prime-field analogue of a version of the entropy power inequality established by Tao over torsion-free groups. Next, we prove an entropy sum-product statement: For independent and identically distributed random variables $X,X'$, the maximum of ${\bf H}(X+X')$ and ${\bf H}(XX')$ is bounded below by a linear combination of the entropy and the min-entropy (R\'enyi entropy of order~$\infty$) of $X$. This result, obtained by bounding entropies of the form ${\bf H}\bigl( X(Y+Z)\bigr)$ from above and below, is valid over arbitrary fields $F$. Over $F={\bf R}$, a slightly stronger inequality is derived. Finally, a weak version of a purely Shannon-entropic sum-product result is developed: If the entropic additive doubling of a random variable $X$ over an arbitrary field is $O(1)$, then its multiplicative doubling is at least proportional to ${\bf H}(X)$.