Frostman random variables, entropy inequalities, and applications

We introduce Frostman conditions for bivariate random variables and study discretized entropy sum-product phenomena in both independent and dependent settings. Fix $0 < s < 1$, and let $(X,Y)$ be a bivariate real random variable with bounded support, whose distribution satisfies a Frostman condition of dimension $s$. Let $φ(x,y)$ be a polynomial obtained from a diagonal polynomial $ρ_1(x)+ρ_2(y)\in \mathbb{R}[x, y]$ of degree $d\ge 2$ by applying an invertible rational linear change of variables in $(x,y)$. We show that there exists $ε= ε(φ,s)>0$ such that $$ \max\{H_n(X+Y), H_n(φ(X,Y))\} \geq n(s+ε) $$ for all sufficiently large $n$, where the precise assumptions on $(X,Y)$ depend on the Frostman level. The proof introduces a novel multi-step entropy framework, combining the submodularity formula, the discretized entropy Balog-Szemerédi-Gowers theorem, and state-of-the-art results on the Falconer distance problem, to reduce general forms to a diagonal core case. As an application, we obtain discretized sum-product type estimates. In particular, for a $δ$-separated set $A\subseteq [0, 1]$ of cardinality $δ^{-s}$, satisfying certain non-concentration conditions, and a dense subset $G\subseteq A\times A$, there exists $ε=ε(s, φ)>0$ such that $$ E_δ(A+_GA) + E_δ(φ_G(A, A)) \ggδ^{-ε}(\#A) $$ for all $δ$ small enough. Here $E_δ(A)$ denotes the $δ$-covering number of $A$, $A+_GA:=\{x+y\colon (x, y)\in G\}$, and $φ_G(A,A):=\{φ(x, y)\colon (x, y)\in G\}$.