A note on maximal operators for the Schr\"{o}dinger equation on $\mathbb{T}^1.$

Motivated by the study of the maximal operator for the Schr\"{o}dinger equation on the one-dimensional torus $\mathbb{T}^1$, it is conjectured that for any complex sequence $\{b_n\}_{n=1}^N$, $$\left\| \sup_{t\in [0,N^2]} \left|\sum_{n=1}^N b_n e \left(x\frac{n}{N} + t\frac{n^2}{N^2} \right) \right| \right\|_{L^4([0,N])} \leq C_\epsilon N^{\epsilon} N^{\frac{1}{2}} \|b_n\|_{\ell^2}$$ 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 $$\left\| \sup_{t\in [0,N^2]} \left|\sum_{n=1}^N b_n e \left(x\frac{n}{N} + ta_n \right) \right| \right\|_{L^4([0,N])} \leq C_\epsilon N^\epsilon N^{\frac{7}{12}} \|b_n\|_{\ell^2}.$$ We further show that this bound is sharp up to a $C_\epsilon N^\epsilon$ factor.