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arXiv · 2507.15196

Entropy Expansion for General Polynomial Images of Frostman Random Variables

Alex Iosevich·Thang Pham·Nguyen Dac Quan·Steven Senger·Boqing Xue
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Abstract

We prove dyadic entropy expansion for the observables $X+Y$ and $f(X,Y)$ under Frostman nonconcentration hypotheses on a prescribed, possibly dependent law. For an integer $n\ge1$, write $H_n(Z)=H(\lfloor 2^nZ\rfloor)$ for the base-two Shannon entropy at resolution $2^{-n}$; thus $n$ indexes the fineness of the dyadic discretization. For every $0 4/3$, we obtain $\max\{H_n(X+Y),H_n(f(X,Y))\}\ge (\frac{s_1+s_2}{2}+\varepsilon)n-O(1)$ for an explicit $\varepsilon>0$. Here the baseline averages the two Frostman exponents. We also classify the exceptional coordinate representations and construct obstructions for algebraic affine directions with Frostman constants uniform in the scale.

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BibTeXRIS

Alex Iosevich, Thang Pham, Nguyen Dac Quan, Steven Senger, Boqing Xue. 2025-07-21. Entropy Expansion for General Polynomial Images of Frostman Random Variables. https://arxiv.org/abs/2507.15196

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