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This paper studies the minimal conditions under which we can establishasymptotic formulae for Waring's problem and other additive problems that maybe tackled by the circle method. We confirm in quantitative terms thewell-known heuristic that a mean value estimate and an estimate of Weyl type,together with suitable distribution properties of the underlying set over a setof admissible residue classes, are sufficient to implement the circle method.This allows us to give a rather general proof of Waring's problem which isapplicable to a range of sufficiently well-behaved thin sets, such as theellipsephic sets recently investigated by the first author.

A minimalist version of the circle method and Diophantine problems over thin sets