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Generalising the recent Kelley–Meka result on sets avoiding arithmetic progressions of length three leads to developments in the theory of the higher energies.
Some new results on the higher energies I
Draft for Geometric and Functional Analysis
We obtain a generalisation of the recent Kelley–Meka result on sets avoiding arithmetic progressions of length three. In our proof we develop the theory of the higher energies. Also, we discuss the case of longer arithmetic progressions, as well as a general family of norms, which includes the higher energies norms and Gowers norms.
Draft for Geometric and Functional Analysis