This is a less ambitious version of [Is the Lebesgue measure of the $x$ so that this exponential sum is $o(\sqrt{N})$ positive?](https://mathoverflow.net/questions/451975/is-the-lebesgue-measure-of-the-x-so-that-this-exponential-sum-is-o-sqrtn) .


Consider $$S\_N(x):=\sum\_{n=1}^N \exp\left(2\pi i\left(\frac12n^2x+\alpha n\right)\right)$$


where $\alpha$ is irrational. If $x$ is an integer then $S\_N(x)$ is bounded for all $N$. Finding rational $x$ shouldn't be hard either. However, is there some pair $x, \alpha$ of irrational numbers so that $S\_N(x)=o(\sqrt N)$?



Does there exist some irrational $x,\alpha$ so that this Weyl sum is $o(\sqrt N)$?
The estimate
$$ S\_N(x) = S\_N(x,\alpha) = o(N^{1/2}) \tag {1}$$
cannot hold when $x$ is irrational.


Heuristically, the reason comes from the (approximate) modularity properties of $S\_N(x,\alpha)$; this expression is periodic mod $1$ in both $x$ and $\alpha$, and from the [van der Corput B-process](https://en.wikipedia.org/wiki/Van_der_Corput%27s_method) (Poisson summation) one morally has a relation
$$ |S\_N(x,\alpha)| 
\sim x^{-1/2} |S\_{Nx}( -1/x, \alpha/x)|$$
when $x$ is non-zero (and $Nx$ is not too small). From periodicity, we see heuristically that if (1) holds for some choice of $x,\alpha,N$, then $x$ (and $\alpha$) can be placed in the interval $[-1/2,1/2]$ while remaining irrational and hence non-zero; from the B-process we may then replace $x,\alpha$ with $-1/x, \alpha/x$ and lower $N$ by a factor of $x$. Iterating this, we morally arrive at the point where (1) would hold in the range $N \sim 1$, which is absurd because the left-hand side jumps by a unit phase when $N$ increments by $1$, while the right-hand side would now be much less than $1$ due to the $o()$ factor.


To make this argument rigorous we introduce the [Jacobi theta function](https://en.wikipedia.org/wiki/Theta_function)
$$ \vartheta(z; \tau) := \sum\_{n \in {\bf Z}} e^{\pi i \tau n^2 + 2\pi i z n}$$
defined for complex $z,\tau$ with $\mathrm{Im} \tau > 0$. Morally we have
$$ S\_N(x,\alpha) \approx \vartheta( \alpha; x + i N^{-1/2} ).$$
We also will need absolute constants
$$ 0 < \delta \lll \varepsilon\_0 \lll \frac{1}{C} \lll 1$$
chosen so that $C$ is sufficiently large, $\varepsilon\_0$ is sufficiently small depending on $C$, and $\delta$ is sufficiently small depending on both $C$ and $\varepsilon\_0$. For any $x,\alpha,\varepsilon$ with $0 < \varepsilon < 1/C$, let $P(x,\alpha,\varepsilon)$ denote the assertion that the estimate
$$ |\vartheta(z; \tau)| \leq \delta (\mathrm{Im} \tau)^{-1/4} (1 + C \mathrm{Im}(\tau))$$
holds whenever $z, \tau$ are complex numbers with
\begin{align\*}
0 < \mathrm{Im}(\tau) &< \varepsilon\\
|\mathrm{Re}(\tau) - x| &< \mathrm{Im}(\tau) (1 - C \mathrm{Im}(\tau))\\
|\mathrm{Im}(z)| &< \mathrm{Im}(\tau)^{1/2}\\
|\mathrm{Re}(z) - \alpha| &< \mathrm{Im}(\tau)^{1/2} (1 - C \mathrm{Im}(\tau)).
\end{align\*}
The $C \mathrm{Im}(\tau)$ terms are technical higher order corrections that we have inserted in order to absorb certain error terms in the iterative component of our analysis, but can be ignored at a first reading.


We then make the following claims:


1. The property $P(x,\alpha,\varepsilon)$ is invariant with respect to shifting $x$ or $\alpha$ by an integer. This is clear from the periodicity properties of $\vartheta$.
2. If (1) holds for some $x, \alpha$, then $P(x,\alpha,\varepsilon)$ holds for all sufficiently small $\varepsilon > 0$. This is a routine exercise in summation by parts, splitting
$ e^{\pi i \tau n^2 + 2\pi i z n}$ into $e^{\pi i x n^2 + 2\pi i \alpha n}$ times a factor which exhibits gaussian decay outside of the region $n = O( (\mathrm{Im} \tau)^{-1/2} )$ and has bounded total variation.
3. If $0 < |x| \leq 1/2$, $|\alpha| \leq 1/2$ and $P(x,\alpha,\varepsilon)$ holds for some $0 < \varepsilon < \varepsilon\_0$, then $P(-1/x, \alpha/x, 2\varepsilon)$ holds. This is a routine computation coming from the Jacobi identity
$$ \vartheta(z;\tau) = (-i\tau)^{-1/2} e^{-\pi i z^2/\tau} \vartheta(z/\tau;-1/\tau)$$
which can be viewed as a special case of the Poisson summation formula.
The main point is that the transformation $(z,\tau) \mapsto (z/\tau, -1/\tau)$ is approximately affine from the region $\tau = -1/x + O(\varepsilon)$, $z = \alpha/x + O(\varepsilon^{1/2})$ to the region $\tau = x + O( \varepsilon x^2 )$, $z = \alpha + O( \varepsilon^{1/2} x)$, up to multiplicative distortion factors of $1 + O( |\tau+1/x|)$ that simplify to $1 - O(\mathrm{Im}(\tau))$ in the region of interest, and which can be tolerated for $C$ large enough.
4. For $\delta$ sufficiently small depending on $\varepsilon\_0, C$, it is not possible for $P(x,\alpha,\varepsilon)$ to hold for any $x,\alpha \in [-1/2,1/2]$ if $\varepsilon$ is in the interval $[\varepsilon\_0, 2\varepsilon\_0]$. For, if this were possible for arbitrarily small $\delta$, then by sending $\delta$ to zero and using compactness we would see that $\vartheta(z,\tau)$ vanished on an open set, hence vanishes for all $z,\tau$ with $\mathrm{Im}(\tau)> 0$ by analytic continuation, which is absurd.


The four properties 1-4 imply that (1) cannot hold when $x$ is irrational. Indeed, if (1) was true for some irrational $x,\alpha$, then by property 2 we have $P(x,\alpha,\varepsilon)$ for some $0 < \varepsilon < \varepsilon\_0$. Using property 1 we can shift $x,\alpha$ to lie in $[-1/2,1/2]$, and then by property 3 we can then double $\varepsilon$ (altering $x,\alpha$ again, but keeping $x$ irrational). Iterating this, we eventually can place $\varepsilon$ in the range $[\varepsilon\_0, 2\varepsilon\_0]$, at which point we contradict property 4.


This argument actually gives the slightly stronger assertion that
$$ \limsup\_{N \to \infty}
N^{-1/2} |\sum\_{n \leq N} e^{\pi i x n^2 + 2 \pi i \alpha n}| \geq c$$
for all irrational $x$, real $\alpha$, and some absolute constant $c>0$.


