Metric theory of Weyl sums

In this talk I'll describe joint work with Chen, Maynard and Shparlinski which obtains new results concerning the Hausdorff dimension of the set of large values of exponential sums. I'll describe two sorts of results:

1) Metric estimates for exponential sums over polynomials: This combines ideas of Cassels with estimates of Hooley for counting solutions to equations with polynomials and shows the set of large values of exponential sums has a Cantor-type structure.

2) Metric estimates for more general smooth functions: We make use of an iterated Van der Corput type oscillatory integral estimate to show a similar Cantor-type structure as in the polynomial case.