  We develop a toolbox for proving decouplings into boxes with diameter smaller
than the canonical scale. As an application of this new technique, we solve
three problems for which earlier methods have failed. We start by verifying the
small cap decoupling for the parabola. Then we find sharp estimates for
exponential sums with small frequency separation on the moment curve in
$\mathbb{R}^3$. This part of the work relies on recent improved Kakeya-type
estimates for planar tubes, as well as on new multilinear incidence bounds for
plates and planks.
  We also combine our method with the recent advance on the reverse square
function estimate, in order to prove small cap decoupling into square-like caps
for the two dimensional cone.
  The Appendix by Roger Heath-Brown contains an application of the new
exponential sum estimates for the moment curve, to the Riemann zeta-function.

Classical Analysis and ODEs