I am interested in the exponential sum


$$\sum\_{n=1}^X \frac{e(c\_1n^2+c\_2 n)}{1-e(c\_1n)}$$


where $c\_2$ is irrational and $e(x)=e^{2\pi i x}$. I know if the denominator is not there, this is a Weyl sum with a lot of bounds.


Is there anything that can be done to relate this sum to a Weyl sum or get some other bounds?


Edit, one thing that might be nice is to get a bound like


$$\left|\sum\_{n=1}^X \frac{e(c\_1n^2+c\_2 n)}{1-e(c\_1n)}\right|\ll \left|\sum\_{n=1}^X e(c\_1n^2+c\_2 n)
\right|\max\_{1\leq n\leq N} \frac{1}{|1-e(c\_1n)|}.$$


Is a bound like this possible?



Exponential sum with weight in bottom
Under your assumption that no $n$ from $1$ to $X$ has $|1 -e(c\_1n)|<\epsilon$, we have an upper bound for your sum of the form $2 \epsilon^{-1} \log X + O(X)$.


This is the "trivial bound" in that it just comes from estimating $$\sum\_{n=1}^X \frac{1}{ |1 - e(c\_1 n )|}.$$


By the pidgeonhole principle, the number of $n$ with $e( c\_1 n)$ at an angle less than $\theta$ from $1$ is at most $ 2 \theta /\epsilon$ since otherwise there would be $2$ such $n$ with angles within an interval of length less than $\epsilon$ and then the absolute value of their difference would have an angle less than $\epsilon$ and thus have $|1 -e(c\_1n) |< \epsilon$.


If $\theta\_1,\dots, \theta\_X$ are any set of angles such that the number with distance less than $\theta$ from $0$ is at most $2\theta/\epsilon$ then


$$ \sum\_{n=1}^{X} \frac{1}{ |1 -e(\theta\_n)|} \leq \sum\_{n=1}^{X} \frac{1}{ |1 -e ( n \epsilon/2))|}$$


because $|1- e(c\_1 n )|$ is an increasing function of the angular difference, and this sum is well-approximated by the integral


$$ 2 \epsilon^{-1} \int\_{\epsilon/2}^{ X \epsilon/2} \frac{ d \theta}{ | 1- e(\theta)|}$$


and since $\frac{1}{ 1- e(\theta)} = \frac{1}{ \theta} + O(1)$ , the integral is bounded by $\log X + O(X \epsilon)$, giving the claim.


Improving this bound by a significant factor seems to require estimates at many different scales, since the integral producing the $\log X$ term has equal contributions from many different scales. I suspect this can be done in such a way that the estimate at each scale is essentially a Weyl sum over a Bohr set, which probably means that cancellation can indeed be found under hypotheses on $c\_1,c\_2$, but one couldn't hope to save more than the $\log X$ factor as one is unlikely to get a bound for the overall sum greater than the bound for a single term.


