# Some (Asymptotic) Bounds on Exponential Sums

Sorry if this is a duplicate, but can't find it anywhere. I'm learning some additive number theory and am stuck on showing some bounds. Let $$S(\alpha, X) = \displaystyle \sum_{x=1}^{X} e(\alpha x)$$ (where $$e(\beta)$$ just means $$e^{2 \pi i \beta}$$). First, I need that $$|S(\alpha, X)| << (2|| \alpha||)^{-1}$$ where $$|| \alpha|| = |\alpha \text{ mod } 1|$$.

Here is what I've gathered: $$S(\alpha, X) = \frac{e(\alpha)-e(\alpha(X+1))}{1-e(\alpha)}$$ by the geometric series formula. Then, the magnitude of this is $$|S(\alpha, X)|^2 = \frac{2-e(\alpha X)-e(-\alpha X)}{2-e(\alpha)-e(-\alpha)} = \frac{2-2\cos(2 \pi \alpha X)}{2-2 \cos (2 \pi \alpha)}$$ I'm not sure where to go from here. I thought I remembered at one point getting an expression with a single $$\sin (2\pi \alpha)$$, which is easy to compare to $$||\alpha||^{-1}$$.

The other bound I'm stuck with is $$\int_{0}^{1} |S(\alpha, X)| \text{ d} \alpha << \log (2X)$$ Again, I'm not really sure how to proceed except with fiddling with the cosine expression I got above, but I have no idea how to relate that to something like $$\log (2X)$$.

For the first bound note that $$2-2 \cos (2 \pi \alpha)=4\sin^2{(\pi \alpha)}$$ and $$4\sin^2{(\pi \alpha)} \ge ||\alpha||^2$$, while the numerator is bounded by $$4$$
For the second bound, we use the first bound and notice that if $$\frac{1}{X} \le \alpha \le 1-\frac{1}{X}$$ we can use the first bound in the integral but for $$\alpha$$ small or near $$1$$ it is better to use the trivial bound $$X$$, so the integral is majorized by:
$$\int_0^{\frac{1}{X}}Xdu+\int_{\frac{1}{X}}^{\frac{1}{2}}\frac{4}{u}du+\int_{\frac{1}{2}}^{1-\frac{1}{X}}\frac{4}{1-u}du+\int_{1-\frac{1}{X}}^{1}Xdu=2+8\log X-8\log 2<< \log X$$