l² decoupling theorem for surfaces in R³

We identify a new way to divide the δ-neighborhood of surfaces M³ into a finitely-overlapping collection of rectangular boxes S. We obtain a sharp (l²,Lᵖ) decoupling estimate using this decomposition, for the sharp range of exponents 2≤ p≤ 4. Our decoupling inequality leads to new exponential sum estimates where the frequencies lie on surfaces which do not contain a line.