Logarithmic joint ergodicity criterion for logarithmic iterates

Let $(X,\mathcal{X},\mu,T_1,T_2)$ be a measure-preserving system. Joint ergodicity for logarithmic averages means convergence of the corresponding logarithmically weighted multiple ergodic averages to the product of the integrals, and ergodicity for logarithmic averages is defined analogously for a sequence of transformations. Logarithmic joint ergodicity criterion. For any such system, the tuple $(T_1^n,T_2^{n+\lfloor\log n\rfloor})_n$ is jointly ergodic for logarithmic averages if and only if $(T_2^{n+\lfloor\log n\rfloor}T_1^{-n})_n$ and $(T_1^n\times T_2^{n+\lfloor\log n\rfloor})_n$ are ergodic for logarithmic averages on $(X,\mathcal{X},\mu)$ and $(X\times X,\mathcal{X}\otimes\mathcal{X},\mu\times\mu)$, respectively. This is a proposed repair of the failure of the corresponding Cesàro-average criterion for logarithmic iterates; its resolution is not supplied in the source.