On an entropy inequality for quadratic forms and applications

2025-07-20

AI Summary

This study addresses the entropy growth problem for non-degenerate rational quadratic forms $varphi(x,y)$ evaluated on independent $(s,C)$-Frostman random variables $X,Y$: we prove the existence of $varepsilon > 0$ and $N in mathbb{N}$ such that for all $n > N$, $max{H_n(X+Y), H_n(varphi(X,Y))} geq n(s+varepsilon)$. Methodologically, we introduce the first multi-scale entropy framework tailored to quadratic forms, integrating submodularity of discrete entropy, a Balog–Szemerédi–Gowers-type theorem, and recent advances on the Falconer distance problem to reduce general $varphi$ to a diagonal core case. Our main contribution is the first rigorous extension of entropy-increment principles from linear (sumset) to nonlinear (quadratic) combinations. As a corollary, for any $delta$-separated set $A$, we obtain a lower bound on the $delta$-covering numbers: $E_delta(A+A) + E_delta(varphi(A,A)) gg delta^{-varepsilon}|A|$.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Machine Learning: Other Foundations of Machine Learning
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Let $φ(x,y)$ 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 $0<s<1$, $C,c_1geq 1$. We prove that there exist a positive constant $ε= ε(s,φ)$ and an integer $N=N(s,C,c_1,φ)$ such that $$maxleft{H_n(X+Y),,H_n(φ(X,Y)) ight} ge n(s+ε)$$ for all $n>N$. 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 derive a result on a discretized sum-product type problem. In particular, for a $δ$-separated set $Asubset [0, 1]$ of cardinality $δ^{-s}$, satisfying some non-concentration conditions, there exists $ε=ε(s, φ)>0$ such that $$E_δ(A+A) + E_δ(φ(A, A)) ggδ^{-ε}(#A) $$ for all $δ$ small enough. Here by $E_δ(A)$ we mean the $δ$-covering number of $A$.

Problem

• Proving entropy inequality for quadratic forms of Frostman variables
• Developing multi-step entropy framework for general forms reduction
• Deriving discretized sum-product result for delta-separated sets

Innovation

• Novel multi-step entropy framework
• Combines submodularity and entropy theorems
• Reduces forms to diagonal core case