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The short version:

Given non-zero real numbers $\alpha$ and $\beta$, can one prove the following estimate in a simple manner? Or does it follow from a well-known result on exponential sums? $$ \sum_{n=1}^N \frac{1}{\sqrt{n}}e(\alpha n + \beta n \log n) = O_{\alpha,\beta}(\log N) $$ (Here $e(x)=e^{2 \pi i x}$) And, if so, can one replace big-O with little-o for certain $\alpha$ and $\beta$?

The background:

This peculiar question came from my recent study of the van der Corput transform (also called Process B or the method of van der Corput). The transform says that given sufficiently "nice" functions $f$ and $g$, with $f$ strictly increasing, $$\sum_{a\le n \le b} g(n) e(f(n)) \approx \sum_{f'(a)\le \nu \le f'(b)} \frac{g(x_\nu)}{\sqrt{f''(x_\nu)}} e(f(x_\nu)-\nu x_\nu +\tfrac{1}{8})$$ where $f'(x_\nu)=\nu$.

The error term in this transformation has been the subject of much study, but the simplest and most traditional is written as $$O( \lambda^{-1/2} + \log(f'(b)-f'(a)+1))$$ where $f''(x) \asymp \lambda$ on $[a,b]$ (for example, Theorem 8.16 in Iwaniec and Kowalski).

Out of curiosity, I considered the case when $g(n) = 1$, $f(n) = k(\tfrac{3}{2})^n$, $a=1$, and $b=x$. If such a sum could be shown to be $o(x)$ for all non-zero integer $k$, then this would imply the equidistribution of the fractional parts of $(\tfrac{3}{2})^n$. The van der Corput transform of such a sum looks like a constant multiple of the sum at the beginning of this problem, with $$\alpha = \frac{1-\log(k\log 3/2)}{\log(3/2)}$$ $$\beta = \frac{1}{\log(3/2)}$$ $$N=k\left(\frac{3}{2}\right)^x \log(3/2)$$

The traditional error terms in this case are, in fact, on the order of $x$, but I believe I have a method to reduce them, which would give the big-O bound at the top of this question as a simple corollary of an exceedingly complicated theorem. I ask this question of mathoverflow because I do not know whether a simpler proof exists, or whether there exists a little-o estimate, which might imply the equidistribution of the fractional parts of $(\tfrac{3}{2})^n$.

I would suspect that any little-o estimates would require very strong conditions on $\alpha$ and $\beta$, given that the sequence $2^n$ does not equidistribute modulo 1, but the van der Corput transforms of the associated exponential sums have very similar $\alpha$ and $\beta$ to the ones mentioned above.

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Second display, $\nu x\nu$ should be $\nu x_{\nu}$? –  Gerry Myerson Jan 23 '12 at 22:20
    
Have you looked through Montgomery's "Ten lectures on the interface between analytic number theory and harmonic analysis"? It might contain similar information to what you've sesn in Iwaniec-Kowalski, but I think it does a thorough job of treating exponential sums of this type. –  Greg Martin Jan 23 '12 at 23:08
    
Joseph, let me be sceptical about the ways you try to achieve the uniform distribution of $\lbrace(3/2)^n\rbrace$: there should be essentially new ingredients in the old techniques. –  Wadim Zudilin Jan 24 '12 at 0:38
    
Gerry Myerson: yes, thank you for catching that. Greg Martin: I have looked through Montgomery's book, but it seems many of the theorems there require the function inside the e( ) to be polynomial in nature and do not react well to the presence of the logarithm. Wadim Zudilin: I am equally skeptical that this would lead to a uniform distribution proof, due to the heavy dependence on properties of alpha and beta, but the question did lead me to looking at this curious sum. –  Joseph Vandehey Jan 24 '12 at 18:21
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