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We present the analytical results for the K -> 3 pi final state interactionsat next-to-leading order (NLO) in CHPT. We also study the recent Cabibbo'sproposal to measure the pi-pi scattering lenghts combination a_0-a_2 from thecusp effect in the pi^0-pi^0 energy spectrum at threshold for K^+ -> pi^0 pi^0pi^+ and K_L -> pi^0 pi^0 pi^0$, and give the relevant formulas to describe itat NLO. For that, we use the NLO CHPT expression to fit the real part of K -> 3pi to data while the pi-pi scattering lenghts are treated non-perturbatively.Using them, we make a quantitative estimate of the theoretical uncertaintity ofthe a_0-a_2 determination at NLO in our approach and obtain that it is notsmaller than 5 % if added quadratically and 7 % if linearly for K^+ -> pi^0pi^0 pi^+. One gets similar theoretical uncertainties if the neutral K_L ->pi^0 pi^0 pi^0 decay data below threshold are used instead. For this decay,there are very large theoretical uncertainties above threshold due tocancellations and data above threshold cannot be used to get the scatteringlenghts. All the numbers we present are in the isospin limit apart of two-pionphase space factors which are physical. We compare our results for the cuspeffect with Cabibbo and Isidori's results and discuss the differences andagreements.  We also comment on the apperance of the singularity at the K -> 3 pipseudo-threshold s=(m_K-m_pi)^2 in the discontinuity that defines the cusp.Comment: 31 pages, 8 figures. v2=v3 Added the full contributions to the cusp  from the real part of the discontinuity. v4 Improved text. Matches published  versio

K→3π final state interactions at NLO in CHPT and Cabibbo’s proposal to measure a0-a2