The circle method is often used to estimate the number of solutions to the equation $$x\_1 + x\_2 + ... x\_k = N$$ if for all $i$ $x\_i\in A\subseteq\mathbb{N}\_0$ and some subset of the nonnegative integers. When we try to use the circle method if $A$ isn't a subset of the nonnegative integers, one runs into issues, since the generating function isn't a polynomial or a fourier series. How would one use the circle method or variants of it to estimate the number of solutions of $$x\_1 + x\_2 + ... x\_k = N$$ if for all $i$, $x\_i\in A$, if $A$ is some subset of the real numbers or complex numbers. 



Circle method on things other than the integers
The circle method is a very powerful and versatile tool which can be made to work in a wide range of situations. It certainly works over number fields, see for example


Skinner - Forms over number fields and weak approximation.


The circle method can even be made to work over function fields, see e.g.


Lee - Forms in many variables over algebraic function fields.


