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.
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1$\begingroup$ What other subsets do you have in mind? $\endgroup$– Yemon ChoiCommented Apr 17, 2014 at 5:19
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$\begingroup$ The Gaussian integers, and subsets of it. $\endgroup$– Mayank PandeyCommented Apr 17, 2014 at 5:48
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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.
answered Apr 17, 2014 at 7:56