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We consider the $(1 + 3)$-dimensional Einstein equations with negativecosmological constant coupled to a spherically-symmetric, massless scalar fieldand study perturbations around the Anti-de Sitter spacetime. We derive theresonant systems, pick out vanishing secular terms and discuss issues relatedto small divisors. Most importantly, we rigorously establish (sharp, in most ofthe cases) asymptotic behaviour for all the interaction coefficients. Thelatter is based on uniform estimates for the eigenfunctions associated to thelinearized operator as well as on some oscillatory integrals.

On the Fourier analysis of the Einstein-Klein-Gordon system: Growth and Decay of the Fourier constants