Entropy lower bounds and sum-product phenomena
This work addresses fundamental problems in information theory and additive combinatorics by providing new entropy inequalities, which are incremental but rigorous extensions of existing results.
The paper establishes lower bounds for the entropy of sums, products, and their combinations over various fields, including deriving a prime-field analogue of an entropy power inequality and proving that for independent and identically distributed random variables, the maximum of the entropies of their sum and product is bounded below by a linear combination of their entropy and min-entropy.
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 (Rényi 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 ${\bf H}(X)$.