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 $\phi (x,y)$ be a polynomial obtained from a diagonal polynomial ${\rho }_1(x)+{\rho }_2(y)\in \mathbb{R}[x,y]$ of degree $d\ge 2$ by applying a change of variables $\mathrm{\Xi }\in G{L}_2(\mathbb{Q})$ in $(x,y)$. We show that there exists $\epsilon =\epsilon (d,\mathrm{\Xi },s)>0$ such that $$\max \{H_n(X+Y),H_n(\phi (X,Y))\}\ge n(s+\epsilon )$$ 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 state-of-the-art results on the Falconer distance problem, a discretized entropy Balog-Szemerédi-Gowers mechanism, and new entropy inequalities adapted to dependent variables, to reduce general polynomials of arbitrary degree to a diagonal quadratic case. As applications, we obtain innovative discretized sum-product type estimates along dense graphs. In particular, for a $\delta$-separated set $A\subseteq [0,1]$ of cardinality ${\delta }^{-s}$, satisfying certain non-concentration conditions, and a dense subset $G\subseteq A\times A$, there exists $\epsilon =\epsilon (s,\phi )>0$ such that $$E_{\delta }(A{+}_{G}A)+E_{\delta }({\phi }_G(A,A))\gg {\delta }^{-\epsilon }(\mathrm{\#}A)$$ for all $\delta$ small enough. Here $E_{\delta }(A)$ denotes the $\delta$-covering number of $A$, $A{+}_{G}A:=\{x+y\text{\hspace{0.17em}⁣}:(x,y)\in G\}$, and ${\phi }_G(A,A):=\{\phi (x,y)\text{\hspace{0.17em}⁣}:(x,y)\in G\}$.