An infinity-pseudometric space has transfinite asymptotic dimension at most omega+n exactly when it admits a two-term integral APD profile of width n+1.
Dranishnikov, Asymptotic topology, Russ
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APD profiles and transfinite asymptotic dimension
An infinity-pseudometric space has transfinite asymptotic dimension at most omega+n exactly when it admits a two-term integral APD profile of width n+1.