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The theory of exponent pairs provides bounds for $$\sum_{N<n<2N} e(f(n)),$$ where f behaves like a monomial. Precise formulations of this are in Graham and Kolesnik (GK) which seems to be what is cited in the literature when one wants to apply the A and B processes.

The problem is that for a function such as $f(n) = n^{3/2}$, it does not satisfy equation 3.3.3 in GK. Nevertheless, one should typically be able to apply the A and B processes.

I am asking for a reference that I can cite which allows one to apply the A and B processes to such functions.

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See Montgomery's book, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, specifically the examples on p.55. He considers variants of your sum, namely $\sum_{n \leq N} e((n/3)^{3/2})$, and $\sum_{n \leq N} e((2n/3)^{3/2})$, and shows that they have very different asymptotic behavior. The former is asymptotically $c N^{3/4}$ while the latter is $O(N^{1/4})$. These examples illustrate why the hypothesis (3.3.3) in [GK] is assumed.

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  • $\begingroup$ Thanks for your response! But, for instance if we consider $e(tn^{3/2})$ and t is transcendental, one should always be able to apply the A and B processes, with some basic assumptions avoiding the case where the derivatives of the resulting amplitude function become too small. Do you know if this is the case? $\endgroup$ – George Shakan Oct 4 '17 at 16:58
  • $\begingroup$ You may certainly apply Poisson summation or the Weyl/van der Corput differencing method to the exponential sum, which you can analyze in an ad-hoc way. The point of the two examples above is that you cannot automatically determine the behavior of the exponential integral only using derivative bounds on the phase function. $\endgroup$ – Matt Young Oct 4 '17 at 17:22
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Applying the A^2B process to n^3/2 will get you to a linear function, for which certainly this method does not work.

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