On an entropy inequality for quadratic forms and applications

We introduce Frostman conditions for bivariate random variables and study discretized entropy sum-product phenomena in both independent and dependent regimes. In the independent setting, we show that for any non-degenerate rational quadratic form $\phi(x,y)$ and $0<s<1$, there exists some positive constant $\epsilon = \epsilon(\phi,s)$ such that any independent marginally $(s,C;s,C)$-Frostman random variable $(X,Y)$ satisfies $$\max\left\{H_n(X+Y),\,H_n(\phi(X,Y))\right\} \ge n(s+\epsilon)$$ for sufficiently large $n$. 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. In the dependent setting, we obtain analogous bounds for special polynomials. As an application, we derive that there exists $\epsilon=\epsilon(s)>0$ such that, for any $\delta$-separated set $A\subset [0, 1]$ of cardinality $\delta^{-s}$ satisfying some non-concentration condition and any subset $G\subseteq A\times A$ with $|G|\geq \delta^\epsilon |A|^2$, the $\delta$-covering numbers satisfy $$ E_\delta(A+_GA) + E_\delta(\alpha \cdot A^{\wedge 2}+_G \beta\cdot A^{\wedge 2}) \gg_{\alpha,\beta,s} C^{-O_s(1)}\delta^{-\epsilon}|A|, $$ whenever $\delta$ is small enough.