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

TL;DR

This paper identifies minimal conditions for applying the circle method to additive problems like Waring's problem, extending its applicability to thin sets with specific distribution properties.

Contribution

It provides a general framework confirming that mean value and Weyl estimates, along with distribution conditions, suffice for the circle method on thin sets.

Findings

Established asymptotic formulas for Waring's problem on thin sets.

Validated the heuristic linking mean value and Weyl estimates to circle method success.

Extended the circle method to new classes of thin sets such as ellipsephic sets.

Abstract

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