Part of the talk will be dedicated to the problem of S.D. Chowla,
concerning the convergence set of the double trigonometric sin-, and
cos-series with the hyperbolic phase. A complete description of the
convergence set, in terms of continued fractions, will be given.
Applications to the density function of the quantum particle in a box will
be discussed. The fractal nature of the density is due to the
self-similarity relations, known in analytic number theory as asymptotic
formulas for H. Weyl's exponential sums on the major arcs. The classical
Gauss' sums play the role of scaling factors, while oscillatory
trigonometric integrals constitute "the pattern" of the arising quantum
carpet.