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 $φ_G(A,A):=\{φ(x, y)\colon (x, y)\in G\}$.

翻译：本文引入了二元随机变量的Frostman条件，并研究了在独立与依赖情形下的离散化熵和积现象。固定$0 < s < 1$，设$(X,Y)$为具有有界支撑的二元实随机变量，其分布满足维度$s$的Frostman条件。令$φ(x,y)$为通过对角多项式$ρ_1(x)+ρ_2(y)\in \mathbb{R}[x, y]$（次数$d\ge 2$）施加可逆有理线性变量变换$(x,y)$所得的多项式。我们证明存在$ε= ε(φ,s)>0$，使得对所有充分大的$n$，有$$ \\max\\{H_n(X+Y), H_n(φ(X,Y))\\} \\geq n(s+ε) $$，其中关于$(X,Y)$的精确假设取决于Frostman水平。证明引入了一种新颖的多步熵框架，结合次模性公式、离散化熵Balog-Szemerédi-Gowers定理，以及Falconer距离问题的最新成果，将一般形式约化至对角核心情形。作为应用，我们得到了离散化和积型估计。特别地，对于满足特定非集中条件的$δ$分离集$A\\subseteq [0, 1]$（基数为$δ^{-s}$）及其稠密子集$G\\subseteq A\\times A$，存在$ε=ε(s, φ)>0$，使得对所有足够小的$δ$，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$φ_G(A,A):=\\{φ(x, y)\\colon (x, y)\\in G\\}$。