Frostman random variables, entropy inequalities, and applications Article Swipe

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Alex Iosevich , Thang Pham , Hồng Quân Nguyễn , Steven Senger , Bing Xue ·

YOU? · · 2025 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2507.15196

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

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Type: preprint
Language: en
Landing Page: http://arxiv.org/abs/2507.15196
PDF: https://arxiv.org/pdf/2507.15196
OA Status: green
OpenAlex ID: https://openalex.org/W4416716172

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DOI: https://doi.org/10.48550/arxiv.2507.15196

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Title: Frostman random variables, entropy inequalities, and applications

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Type: preprint

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Language: en

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Publication year: 2025

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Publication date: 2025-07-21

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Authors: Alex Iosevich, Thang Pham, Hồng Quân Nguyễn, Steven Senger, Bing Xue

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Landing page: https://arxiv.org/abs/2507.15196

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