Equivalence between sofic metric mean dimension and sofic $p$-metric mean dimension with a product formula

Abstract

In this paper, we prove the equivalence between sofic $p$-metric mean dimension and sofic metric mean dimension. This answers a question of Hayes in \cite{HB }. Furthermore, we establish the product formula for the sofic $p$-metric mean dimension.

A critical and irreparable error has been discovered in the proof of the paper's central theorem. This error fundamentally undermines the primary conclusion of the work. Given the severity of this issue, I believe it is necessary to withdraw the paper entirely to avoid disseminating incorrect results