Implications for large sums of roots of unity

Question

I have some coefficients $(a_n)_{n \leq N} \subset \mathbb{R}$ such that $a_n \geq 0$ and their average value is one, i.e. $\frac{1}{N} \sum_{n \leq N} a_n = 1$. Suppose that $$ \Bigl| \sum_{n \leq N} a_n e(n/N) \Bigr| > N^{\varepsilon}, $$ where $e(x):= e^{2\pi i x}$. I want to make some deductions about the coefficients $(a_n)$. A simple example is that the largest $a_n$ must be $\geq 1$, which follows purely from the average result and the pigeonhole principle. We can't improve on this much using the above inequality, since it's possible to have $a_n = 2$ for $n \leq N/2$, $a_n = 0$ otherwise, and the sum is $\gg N$. However, such an example imparts a lot of structure on the coefficients.

Deriving a quantitative statement about the structure of $a_n$ would be very useful. Any references to where an inverse problem of this type has been studied before would also be greatly appreciated. Even a statement like there must be an $a_n$ larger than $1 + f(N, \varepsilon)$ may be of use.

Answer 1

It depends on what range of $\epsilon$ you are interested in. If the exponential sum is near its max possible size ($> \delta N$) then you are saying that your sequence correlates with an exponential character $e(\cdot)$ and you can get a density increment on a sub-progression (al la the standard density increment proof of Roth's theorem). If you further have that the sum correlates with a large number of characters then you deduce some further additive structure (see for instance, Additive combinatorics and large Fourier coefficients and A question regarding Bourgain's paper on $\Lambda(p)$-subsets) for $\epsilon \geq 1/2$. Generic coefficients (such as randomly chosen sequences of 0's and 1's) will give a sequence where your sum is near $N^{1/2}$, so you can't say much below that.