TITLE: Is $f(x,y)=\sum_{n\in\mathbb{Z}\backslash\{0\}}\frac{1}{n}e^{2\pi i(xn+yn^2)}$ essentially bounded?
QUESTION [8 upvotes]: Let $$f(x,y)=\sum_{n\in\mathbb{Z}\backslash\{0\}}\frac{1}{n}e^{2\pi i(xn+yn^2)}
$$
Is it true that $\|f\|_{L^{\infty}(\mathbb{R}^2)}<\infty$? i.e. is $f$ essentially bounded?

REPLY [5 votes]: Prof. Tao's answer is excellent. I also found two research papers answering the question so I list them below as complementary reference:

G.I.Arkhipov and K.I.Oskolkov, On a special trigonometric series and its applications, 1989 Math. USSR Sb. 62 145
Link to the article:
http://iopscience.iop.org/0025-5734/62/1/A10
See Theorem 1.
E.S.Stein and S.Wainger, Discrete analogues of singular Radon transform, Bulletin of AMS 1990
Link to the article: http://www.ams.org/journals/bull/1990-23-02/S0273-0979-1990-15973-7/S0273-0979-1990-15973-7.pdf
See the Lemma in Section 6

The key of their results is that the upper bound depends only on the degree (not the coefficients, i.e. $x$ and $y$ in the question) of the polynomial.