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Characterizing Small Circuit Classes from FAC^0 to FAC^1 via Discrete Ordinary Differential Equations
Melissa Antonelli, Arnaud Durand, Juha Kontinen
https://arxiv.org/abs/2506.23404

Characterizing Small Circuit Classes from FAC^0 to FAC^1 via Discrete Ordinary Differential Equations
In this paper, we provide a uniform framework for investigating small circuit classes and bounds through the lens of ordinary differential equations (ODEs). Following an approach recently introduced to capture the class of polynomial-time computable functions via ODE-based recursion schemas and later applied to the context of functions computed by unbounded fan-in circuits of constant depth (FAC^0), we study multiple relevant small circuit classes. In particular, we show that natural restrictio…

Entropic additive energy and entropy inequalities for sums and products
Rupert Li, Lampros Gavalakis, Ioannis Kontoyiannis
https://arxiv.org/abs/2506.20813

Entropic additive energy and entropy inequalities for sums and products
Following a growing number of studies that, over the past 15 years, have established entropy inequalities via ideas and tools from additive combinatorics, in this work we obtain a number of new bounds for the differential entropy of sums, products, and sum-product combinations of continuous random variables. Partly motivated by recent work by Goh on the discrete entropic version of the notion of "additive energy", we introduce the additive energy of pairs of continuous random variables and prov…