A solution to the multidimensional additive homological equation

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Abstract

We prove that, for a finite-dimensional real normed space V, every bounded mean zero function f ∈ L([0, 1]; V) can be written in the form f = g ◦ T − g for some g ∈ L([0, 1]; V) and some ergodic invertible measure preserving transformation T of [0, 1]. Our method moreover allows us to choose g, for any given ε > 0, to be such that ∥g∥ ⩽ (SV + ε)∥f∥, where SV is the Steinitz constant corresponding to V.