Quantitative Preservations of Ulam Stability-type Estimates
Mason Sharp
TL;DR
The paper addresses preserving quantitative Ulam-stability-type estimates when extending amenable groups, using integral-transform techniques and Hölder-type decay assumptions to obtain explicit bounds. It proves a main product-preservation result: if $G$ is amenable and $H$ is SUS with $F_H(\varepsilon)=O(\varepsilon)$, then $G\times H$ inherits SUS with a matching rate, with extensions to $F_H(\varepsilon)=O(\varepsilon^s)$ for $s>\tfrac{1}{2}$ and to certain semidirect products. It also develops inductivity results in finite dimensions and shows how to realize existential closures with controlled stability moduli, culminating in conjectures that SUS moduli decay linearly at zero. The work provides a quantitative framework for stability under natural group-constructs, with potential implications for understanding whether nonamenable SUS groups exist and for constructing closures with uniform stability bounds.
Abstract
We show some preservation results of amenably extending strongly Ulam stable groups under mild decay assumptions, including quantitative preservation of asymptotic bounds under the assumption that the modulus of stability is Hölder continuous of exponent $s>\frac 1 2$ at 0, utilizing some simplistic integral estimates. Additionally, we show some partial results around inductive preservation of modulus bounds in infinite dimensions using these integral estimates, as well as strong quantitative preservation in the finite dimensional case. This implies the existence of $\mathfrak{U}$ uniformly stable existential closures among groups with sufficiently large Lipschitz estimates of any countable group. Finally, we show quantitative control preserving difficulty of approximation of maps over stable groups on diagonally embedding into higher dimensions.