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Abstract

We introduce Frostman conditions for bivariate random variables and study discretized entropy sum-product phenomena in both independent and dependent regimes. We prove that for any non-degenerate rational quadratic form $\phi(x, y)$ and parameter $0 < s < 1$, there exists a positive constant $\epsilon = \epsilon(\phi,s)$ such that \[ \max\{H_n(X+Y), H_n(\phi(X,Y))\} \geq n(s+\epsilon) \] for sufficiently large $n$, where the precise conditions 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-Szemer\'{e}di-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 derive a result on a discretized sum-product type problem. In particular, for a $\delta$-separated set $A\subset [0, 1]$ of cardinality $\delta^{-s}$, satisfying some non-concentration conditions, and a dense subset $G\subset A\times A$, 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 by $E_\delta(A)$ we mean the $\delta$-covering number of $A$, $\{A+_GA:=\{x+y\colon (x, y)\in G\}$, and $\phi_G(A):=\{\phi(x, y)\colon (x, y)\in G\}$.