Frostman random variables, entropy inequalities, and applications

Abstract

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 $\ddot{I}•(x,y)$ be a polynomial obtained from a diagonal polynomial $\ddot{I}{}_1(x)+\ddot{I}{}_2(y)\in \mathbb{R}[x,y]$ of degree $d\ge 2$ by applying a change of variables $\hat{I}ž\in G{L}_2(\mathbb{Q})$ in $(x,y)$. We show that there exists $\hat{I}\mu =\hat{I}\mu (d,\hat{I}ž,s)>0$ 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 $\hat{I}´$-separated set $A\subseteq [0,1]$ of cardinality $\hat{I}{´}^{-s}$, satisfying certain non-concentration conditions, and a dense subset $G\subseteq A\times A$, there exists $\hat{I}\mu =\hat{I}\mu (s,\ddot{I}•)>0$ such that $$E_{\hat{I}}´(A{+}_{G}A)+E_{\hat{I}}´(\ddot{I}{•}_G(A,A))\gg \hat{I}{´}^{-\hat{I}\mu }(\#A)$$ for all $\hat{I}´$ small enough. Here $E_{\hat{I}}´(A)$ denotes the $\hat{I}´$-covering number of $A$, $A{+}_{G}A:=\{x+y:(x,y)\in G\}$, and $\ddot{I}{•}_G(A,A):=\{\ddot{I}•(x,y):(x,y)\in G\}$.

v5 (92 pages). Changes relative to v4: (i) Theorem 1.3 is improved by removing the H_n(X, Y)/2 term (hence it implies Theorem 1.4). (ii) Added a subsection on the main ideas and novelties. (iii) Added applications to sum-product type questions for adaptable (discrete) sets. (iv) Corrected typos and minor errors