  In 2017, Benatar and Maffucci arXiv:1708.07015 established an asymptotic law
for the variance of the nodal surface of arithmetic random waves on the 3-torus
in the high-energy limit. In a subsequent work, Cammarota arXiv:1708.07679
proved a universal non-Gaussian limit theorem for the nodal surface. In this
paper, we study the nodal intersection length and the number of nodal
intersection points associated, respectively, with two and three independent
arithmetic random waves of same frequency on the 3-torus. For these quantities,
we compute their expected value, asymptotic variance as well as their limiting
distribution. Our results are based on Wiener-It\^{o} expansions for the volume
and naturally complement the findings of Cammarota arXiv:1708.07679. At the
core of our analysis lies an abstract cancellation phenomenon applicable to the
study of level sets of arbitrary Gaussian random fields, that we believe has
independent interest.
