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 a change of variables $Ξ\in GL_2(\mathbb{Q})$ in $(x,y)$. We show that there exists $ε= ε(d,Ξ,s)>0$ 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 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 $δ$-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 $φ_G(A,A):=\{φ(x, y)\colon (x, y)\in G\}$.

翻译：本文引入了二元随机变量的弗罗斯特曼条件，并研究了在独立与相依两种设定下的离散化熵和积现象。固定 $0 < s < 1$，令 $(X,Y)$ 为具有有界支撑的二元实随机变量，其分布满足维度为 $s$ 的弗罗斯特曼条件。令 $φ(x,y)$ 为通过对角多项式 $ρ_1(x)+ρ_2(y)\in \mathbb{R}[x, y]$（其次数 $d\ge 2$）在变量 $(x,y)$ 上应用变换 $Ξ\in GL_2(\mathbb{Q})$ 得到的多项式。我们证明存在 $ε= ε(d,Ξ,s)>0$，使得对于所有充分大的 $n$，有 \[ \max\{H_n(X+Y), H_n(φ(X,Y))\} \geq n(s+ε) \]，其中对 $(X,Y)$ 的精确假设取决于弗罗斯特曼水平。证明引入了一种新颖的多步熵框架，结合了关于法尔科纳距离问题的最新成果、一个离散化熵巴洛格-塞迈雷迪-高尔斯机制，以及适用于相依变量的新熵不等式，从而将任意次数的多项式约化到对角二次情形。作为应用，我们得到了沿稠密图的创新性离散化和积型估计。特别地，对于一个基数满足 $\#A = δ^{-s}$、满足特定非集中性条件的 $δ$-分离集 $A\subseteq [0, 1]$，以及一个稠密子集 $G\subseteq A\times A$，存在 $ε=ε(s, φ)>0$，使得对于所有足够小的 $δ$，有 $$E_δ(A+_GA) + E_δ(φ_G(A, A)) \ggδ^{-ε}(\#A) $$。此处 $E_δ(A)$ 表示 $A$ 的 $δ$-覆盖数，$A+_GA:=\{x+y\colon (x, y)\in G\}$，且 $φ_G(A,A):=\{φ(x, y)\colon (x, y)\in G\}$。