A standard Transformer with O(ε^{-d0/α}) blocks can approximate any bounded d0-dimensional Hölder function of smoothness α to accuracy ε, but at least Ω(ε^{-d0/(4α)}) blocks are required.
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Approximation Error Upper and Lower Bounds for H\"{o}lder Class with Transformers
A standard Transformer with O(ε^{-d0/α}) blocks can approximate any bounded d0-dimensional Hölder function of smoothness α to accuracy ε, but at least Ω(ε^{-d0/(4α)}) blocks are required.