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arXiv · 1312.4639

Finite forms of Gowers' Theorem on the oscillation stability of $c_0$

Diana Ojeda-Aristizabal
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Abstract

We give a constructive proof of the finite version of Gowers' $FIN_k$ Theorem and analyse the corresponding upper bounds. The $FIN_k$ Theorem is closely related to the oscillation stability of $c_0$. The stabilization of Lipschitz functions on arbitrary finite dimensional Banach spaces was studied well before by V. Milman. We compare the finite $FIN_k$ Theorem with the finite stabilization principle in the case of spaces of the form $\ell_{\infty}^n$, $n\in\mathbb{N}$ and establish a much slower growing upper bound for the finite stabilization principle in this particular case.

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Diana Ojeda-Aristizabal. 2013-12-17. Finite forms of Gowers' Theorem on the oscillation stability of $c_0$. https://arxiv.org/abs/1312.4639

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