In 1998, Gowers proved the first effective bounds for Szemerédi's theorem, showing that any subset formula_1 free of "k"-term arithmetic progressions has cardinality formula_2 for an appropriate formula_3. One of the ingredients in Gowers's argument is a tool now known as the Balog–Szemerédi–Gowers theorem, which has found many further applications. He also introduced the Gowers norms, a tool in arithmetic combinatorics, and provided the basic techniques for analysing them. This work was further developed by Ben Green and Terence Tao, leading to the Green–Tao theorem.