Finite forms of Gowers' Theorem on the oscillation stability of c₀

الأصل · EN

We give a constructive proof of the finite version of Gowers' FINₖ Theorem and analyse the corresponding upper bounds. The FINₖ Theorem is closely related to the oscillation stability of c₀. The stabilization of Lipschitz functions on arbitrary finite dimensional Banach spaces was studied well before by V. Milman. We compare the finite FINₖ Theorem with the finite stabilization principle in the case of spaces of the form ℓ∞ⁿ, n and establish a much slower growing upper bound for the finite stabilization principle in this particular case.