A note on maximal operators for the Schrödinger equation on ${\mathbb{T}}^1\mathrm{.}$

Motivated by the study of the maximal operator for the Schrödinger equation on the one-dimensional torus ${\mathbb{T}}^1$, it is conjectured that for any complex sequence $\{b_n{\}}_{n=1}^N$, $${\parallel \underset{t\in [0,N^2]}{\sup }\mid \sum _{n=1}^N{b}_{n}e(x\frac{n}{N}+t\frac{n^2}{N^2})\mid \parallel }_{L^4([0,N])}\le C_{\epsilon }{N}^{\epsilon }{N}^{\frac{1}{2}}\mathrm{\parallel }{b}_n{\mathrm{\parallel }}_{{\mathrm{\ell }}^2}$$ In this note, we show that if we replace the sequence $\{\frac{n^2}{N^2}{\}}_{n=1}^N$ by an arbitrary sequence $\{a_n{\}}_{n=1}^N$ with only some convex properties, then $${\parallel \underset{t\in [0,N^2]}{\sup }\mid \sum _{n=1}^N{b}_{n}e(x\frac{n}{N}+t{a}_n)\mid \parallel }_{L^4([0,N])}\le C_{\epsilon }{N}^{\epsilon }{N}^{\frac{7}{12}}\mathrm{\parallel }{b}_n{\mathrm{\parallel }}_{{\mathrm{\ell }}^2}\mathrm{.}$$ We further show that this bound is sharp up to a $C_{\epsilon }{N}^{\epsilon }$ factor.