pointwise decay estimate to pass to arbitrarily long times. The difficulty of this process depends on the dimension (with higher dimensional cases being generally easier), and on the exact combination of speeds $c_{\alpha}$ and masses $b_{\alpha}$ (which determines the structure of the so-called “resonance set”). The easiest case is a single Klein-Gordon equation in $3D$, and was first settled independently by Klainerman [@Kl85] and Shatah [@S85] (see also Guo [@Gu98] for the Euler-Poisson system, which linearizes to Klein-Gordon). In the case of (\[nlkg\]) with the *same* speed ($c_{\alpha}=1$) and multiple masses, Hayashi-Naumkin-Wibowo [@HNW] obtained global existence in $3D$, and Delort-Fang-Xue [@DFX] proved the same result in $2D$ under a non-resonance assumption. Note that the above works are more or less based on physical space analysis.

The case of (\[nlkg\]) with *multiple* speeds, even in $3D$, is significantly more challenging. In recent years there have been works developing new, Fourier-based techniques to solve these problems (see a brief discussion in Section \[strategykg\] below). These include for example Germain [@G11] where a specific case of (\[nlkg\]) with the same mass was considered in $3D$, and Ionescu-Pausader [@IP12], where the general system (\[nlkg\]) was studied in $3D$, with speeds and masses $(b_{\alpha},c_{\alpha})$ satisfying 