Very good point @imakhlin, may be each term is bounded independently of $r$. I also agree that the shape of the ball with which we intersect is cumbersome and kind of artificial. Could you point out to me this paper from Brion that you mentioned. I would be happy to learn more about the technique he is using.

should probably replaced the denominator $(1-\alpha)$ by $\min\{\alpha,1-\alpha\}$.

You can use for example the Euler-Maclaurin formula (up to order 2) to show that the difference between the exponential sum (so here over a consecutive set of integers) and the corresponding integral is bounded. The bound only depends on $0<\alpha=\alpha_1<1$ and looking at my calculation it looks like $1+\pi \alpha/6+\pi/3\cdot\alpha^2/(1-\alpha)$ which is independent of the length of the sum.

I should also add just to be clear that it is not because some math research is easy to explain to the layman or has sexy applications in the real world that automatically it is better (or more desirable) than some very esoteric research with no foreseeable applications . The real world is just way too much complicated to adopt such a naive point of view. Fixing a given set of values and goals S, I can easily come up with examples of math research A that I would consider less valuable than some more esoteric research B according to S.

I agree generally with Shulman's opinion on the joint articulation of teaching and research. But the interesting question here regarding research (a difficult one where some disagreement is likely to take place which makes it potentially interesting) is how much resources should we (society) be devoting for this collective enterprise. Resources are for sure limited, once you choose to use some resources on A rather than B then quite often, in the real world, there is a trade-off in the style that X will benefit from this choice while Y not (and sometimes can even be at the expense of Y).

Thanks @Asaf, I just corrected the obvious typo, there was indeed no $t$ there.