Discrepancy and Eigenvalues of Cayley Graphs

We consider quasirandom properties for Cayley graphs of finite abelian groups. We show that having uniform edge-distribution (i.e., small discrepancy) and having large eigenvalue gap are equivalent properties for such Cayley graphs, even if they are sparse. This positively answers a question of Chung and Graham ["Sparse quasi-random graphs", Combinatorica 22 (2002), no. 2, 217-244] for the particular case of Cayley graphs of abelian groups, while in general the answer is negative.
Comment: Dedicated to the memory of Professor Miroslav Fiedler, 33 pages, second version addresses changes arising from the referee report and includes an appendix with an earlier argument