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Given a set $A$ of positive integers, set $r_{A,h}(N)$ to be the number of $h$-tuples $(a_1, \cdots, a_h)$ such that $N = a_1 + \cdots + a_h$. Set $f(z) = \sum_{a \in A} z^a$. Then by Cauchy's Theorem we have that $r_{A,h}(N) = \frac{1}{2\pi i} \int_{|z| = r} \frac{f(z)^h}{z^{N+1}}dz$ for $0 < r < 1$. This can be simplified if we replace $f$ with the trigonometric polynomial $f_N(z) = \sum_{a \in A, a \leq N} z^a$. The key is to estimate the integral, and to do this one breaks the circle into major and minor arcs and to show that the integral over the minor arcs is negligible compared to the integral over the major arc. A key component is to show that the integral over the major arc can be factored into a singular integral, usually denoted $J(N)$ or $J_A(N)$ if the set $A$ is unclear, and a 'singular series' $\mathfrak{S}(N)$ or $\mathfrak{S}_A(N)$ with some small error term. The Circle Method has been applied fruitfully to provide an asymptotic formula for the classical Waring's problem and Vinogradov's Theorem giving an asymptotic formula for sum of three primes (see for example Nathanson's Additive Number Theory - Classical Bases).

My question is, if $A$ is an arbitrary set of positive integers, how does one go about constructing $\mathfrak{S}_A(N)$ or $J_A(N)$? If this is impossible in general or far too intractable, what properties of the integer powers (less interesting) and the primes (more interesting) allows us to find explicit singular integrals and singular series? Can anyone give an example of an 'arbitrary' (meaning not given by a specific formula, somewhat like the primes) set of positive integers with a nice singular series and a singular integral?

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up vote 2 down vote accepted

$A$ should have a fairly regular distribution in arithmetic progressions to yield a reasonable singular series and singular integral. This is why Vinogradov published his solution to the ternary Goldbach problem only after the Siegel-Walfisz theorem was available. The idea is that the relevant trigonometric polynomial has its peak at $z=1$ (obviously) and similar looking but smaller peaks at roots of unity (which is not obvious at all). Now the extent of success depends in large on the extent of "similarity" in this statement.

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Is there a non-trivial example of a base that is not the primes or the integer powers such that it is possible to obtain an asymptotic of some sort? –  Stanley Yao Xiao Jan 12 '11 at 5:41
    
There are many variants of the original Waring problem that have been studied with the circle method, e.g. polynomial equations with restricted variables (primes, almost primes, square-free numbers, smooth numbers). In the case when some modularity works in the background, one can obtain stronger results (e.g. asymptotics for the partition function, Kloosterman's refinement etc.) I guess the lack of variety of examples is not a limitation of the circle method, but rather a limitation of concrete additive problems we are interested in. Somehow we find primes and powers especially beautiful. –  GH from MO Jan 12 '11 at 13:46
    
There have also been a few negative theorems proved by related techniques such as the Erdős-Fuchs theorem (Google finds it). –  GH from MO Jan 12 '11 at 13:48
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