乌拉姆稳定性型估计的定量保持
Quantitative Preservations of Ulam Stability-type Estimates
AI总结：
该研究在温和衰减假设下，通过积分估计展示了强乌拉姆稳定群顺从扩张的定量保持结果，包括无限维部分归纳保持、有限维强定量保持，还证明了可数群存在℘一致稳定存在闭包及高维嵌入时逼近难度的定量控制保持。
摘要：
我们在温和衰减假设下展示了强乌拉姆稳定群的顺从扩张的一些保持结果，包括在稳定性模在0点处是指数s>1/2的Hölder连续的假设下，利用一些简单的积分估计对渐近界的定量保持。此外，我们利用这些积分估计展示了无限维中模界归纳保持的部分结果，以及有限维情形下的强定量保持。这意味着在具有足够大Lipschitz估计的群中，任何可数群都存在℘一致稳定的存在闭包。最后，我们展示了在对角嵌入到更高维时，稳定群上映射的逼近难度的定量控制保持。
英文摘要：
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.
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