We consider doubly-periodic travelling waves at the surface of an infinitely
deep perfect fluid, only subjected to gravity g and resulting from the
nonlinear interaction of two simply periodic travelling waves making an angle
2θ between them. \newline Denoting by μ=gL/c2 the dimensionless
bifurcation parameter (L is the wave length along the direction of the
travelling wave and c is the velocity of the wave), bifurcation occurs for
μ=cosθ. For non-resonant cases, we first give a large family of
formal three-dimensional gravity travelling waves, in the form of an expansion
in powers of the amplitudes of two basic travelling waves. "Diamond waves" are
a particular case of such waves, when they are symmetric with respect to the
direction of propagation.\newline \emph{The main object of the paper is the
proof of existence} of such symmetric waves having the above mentioned
asymptotic expansion. Due to the \emph{occurence of small divisors}, the main
difficulty is the inversion of the linearized operator at a non trivial point,
for applying the Nash Moser theorem. This operator is the sum of a second order
differentiation along a certain direction, and an integro-differential operator
of first order, both depending periodically of coordinates. It is shown that
for almost all angles θ, the 3-dimensional travelling waves bifurcate
for a set of "good" values of the bifurcation parameter having asymptotically a
full measure near the bifurcation curve in the parameter plane (θ,μ).Comment: 119