In this talk, we test M. Berry's ansatz on nodal deficiency in presence of boundary. The square billiard is studied, where the high spectral degeneracies allow for the introduction of a Gaussian ensemble of random Laplace eigenfunctions ("boundary-adapted arithmetic random waves"). As a result of a precise asymptotic analysis, two terms in the asymptotic expansion of the expected nodal length are derived, in the high energy limit along a generic sequence of energy levels.
In particular, we shall focus on a number-theoretic aspect of this problem, describing the techniques introduced by E. Bombieri and J. Bourgain to study additive equations for integral points on the circles. This is based on a joint work with V. Cammarota and I. Wigman.
