On an entropy inequality for quadratic forms and applications

Let $\phi(x,y)$ be a non-degenerate rational quadratic form. Let $X$ and $Y$ be independent $(s, C)$-Frostman random variables whose ranges are contained in $[-c_1, c_1]$, with $0<s<1$, $C,c_1\geq 1$. We prove that there exist a positive constant $\epsilon = \epsilon(s,\phi)$ and an integer $N=N(s,C,c_1,\phi)$ such that $$\max\left\{H_n(X+Y),\,H_n(\phi(X,Y))\right\} \ge n(s+\epsilon)$$ for all $n>N$. 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. 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, there exists $\epsilon=\epsilon(s, \phi)>0$ such that $$E_\delta(A+A) + E_\delta(\phi(A, A)) \gg\delta^{-\epsilon}(\#A) $$ for all $\delta$ small enough. Here by $E_\delta(A)$ we mean the $\delta$-covering number of $A$.