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 $\Xi\in GL_2(\mathbb{Q})$ in $(x,y)$. We show that there exists $\epsilon = \epsilon(d,\Xi,s)>0$ such that $$\max\{H_n(X+Y), H_n(\phi(X,Y))\} \geq n(s+\epsilon)$$ 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-Szemer\'{e}di-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 $$E_\delta(A+_GA) + E_\delta(\phi_G(A, A)) \gg\delta^{-\epsilon}(\#A)$$ for all $\delta$ small enough. Here $E_\delta(A)$ denotes the $\delta$-covering number of $A$, $A+_GA:=\{x+y\colon (x, y)\in G\}$, and $\phi_G(A,A):=\{\phi(x, y)\colon (x, y)\in G\}$.