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We study the splitting of invariant manifolds of whiskered (hyperbolic) toriwith three frequencies in a nearly-integrable Hamiltonian system, whosehyperbolic part is given by a pendulum. We consider a 3-dimensional torus witha fast frequency vector $\omega/\sqrt\varepsilon$, with$\omega=(1,\Omega,\widetilde\Omega)$ where $\Omega$ is a cubic irrationalnumber whose two conjugates are complex, and the components of $\omega$generate the field $\mathbb Q(\Omega)$. A paradigmatic case is the cubic goldenvector, given by the (real) number $\Omega$ satisfying $\Omega^3=1-\Omega$, and$\widetilde\Omega=\Omega^2$. For such 3-dimensional frequency vectors, thestandard theory of continued fractions cannot be applied, so we develop amethodology for determining the behavior of the small divisors $\langlek,\omega\rangle$, $k\in{\mathbb Z}^3$. Applying the Poincar\'e-Melnikov method,this allows us to carry out a careful study of the dominant harmonic (whichdepends on $\varepsilon$) of the Melnikov function, obtaining an asymptoticestimate for the maximal splitting distance, which is exponentially small in$\varepsilon$, and valid for all sufficiently small values of~$\varepsilon$.This estimate behaves like $\exp\{-h_1(\varepsilon)/\varepsilon^{1/6}\}$ and weprovide, for the first time in a system with 3 frequencies, an accuratedescription of the (positive) function $h_1(\varepsilon)$ in the numerator ofthe exponent, showing that it can be explicitly constructed from the resonanceproperties of the frequency vector $\omega$, and proving that it is aquasiperiodic function (and not periodic) with respect to $\ln\varepsilon$. Inthis way, we emphasize the strong dependence of the estimates for the splittingon the arithmetic properties of the frequencies.

Exponentially small splitting of separatrices associated to 3D whiskered tori with cubic frequencies