Frostman random variables, entropy inequalities, and applications
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| Format: | Preprint |
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2025
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| _version_ | 1866915826396823552 |
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| author | Iosevich, Alex Pham, Thang Quan, Nguyen Dac Senger, Steven Xue, Boqing |
| author_facet | Iosevich, Alex Pham, Thang Quan, Nguyen Dac Senger, Steven Xue, Boqing |
| contents | 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 a change of variables $Ξ\in GL_2(\mathbb{Q})$ in $(x,y)$. We show that there exists $ε= ε(d,Ξ,s)>0$ 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 $δ$-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 $ϕ_G(A,A):=\{ϕ(x, y)\colon (x, y)\in G\}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_15196 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Frostman random variables, entropy inequalities, and applications Iosevich, Alex Pham, Thang Quan, Nguyen Dac Senger, Steven Xue, Boqing Classical Analysis and ODEs Information Theory Combinatorics 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 a change of variables $Ξ\in GL_2(\mathbb{Q})$ in $(x,y)$. We show that there exists $ε= ε(d,Ξ,s)>0$ 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 $δ$-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 $ϕ_G(A,A):=\{ϕ(x, y)\colon (x, y)\in G\}$. |
| title | Frostman random variables, entropy inequalities, and applications |
| topic | Classical Analysis and ODEs Information Theory Combinatorics |
| url | https://arxiv.org/abs/2507.15196 |