The fourth volume of Stein & Shakarchi's series on analysis has a nice account of the application of harmonic analysis to lattice point problems (e.g. Gauss' circle problem and the Dirichlet divisor problem).

Martin Huxley's "Area, Lattice Points, and Exponential Sums" is a more thorough (but very readable) account of the business of counting lattice points.

My advisor (Alex Iosevich) has done work (with others) in this area, using some more recent techniques in harmonic analysis (or as Alex would say "using nuclear weapons on small animals"). Anyway, it's still an active area, though as suggested above, you might want to work on whatever is popular in your department.

Also, Vinogradov's "Elements of Number Theory" has a lot of nice exercises on exponential sums. Recently, techniques from additive combinatorics have led to new results on exponential sums; Bourgain wrote a nice survey called "Sum-product Theorems and Applications".