  We show that for some constant $\beta > 0$, any subset $A$ of integers
$\{1,\ldots,N\}$ of size at least $2^{-O((\log N)^\beta)} \cdot N$ contains a
non-trivial three-term arithmetic progression. Previously, three-term
arithmetic progressions were known to exist only for sets of size at least
$N/(\log N)^{1 + c}$ for a constant $c > 0$.
  Our approach is first to develop new analytic techniques for addressing some
related questions in the finite-field setting and then to apply some analogous
variants of these same techniques, suitably adapted for the more complicated
setting of integers.
