Generic Strichartz conjecture for the Schrödinger equation on irrational tori

Let $d\geq 2$ and let $(\beta_1,\dots,\beta_d)\in[1,2]^d$ be generic, meaning outside a Lebesgue-null set. Define

$$\Delta_\beta=\frac{1}{2\pi}\left(\beta_1\partial_1^2+\cdots+\beta_d\partial_d^2\right)$$

on the square torus $\mathbb{T}^d=[0,1]^d$. For $N\geq 1$, $T\geq 1$, and $f$ with Fourier support in the Euclidean ball $B(0,N)$, let

$$\theta(p)=\begin{cases}0,&p\in\left[1,\frac{2(d+2)}{d}\right),\\[2pt]\frac d2\left(p-\frac{2(d+2)}d\right),&p\in\left[\frac{2(d+2)}d,6\right),\\[2pt]2d-2,&p\in[6,\infty). \end{cases}$$

Generic Strichartz conjecture. For arbitrarily small $\varepsilon>0$, one has

$$\left\|e^{it\Delta_\beta}f\right\|_{L^p([0,T]\times\mathbb{T}^d)}\lesssim_{\varepsilon}N^{\varepsilon}\left(1+N^{\frac d2-\frac{d+2}{p}}\right)\left[1+\left(\frac{T}{N^{\theta(p)}}\right)^{1/p}\right]\|f\|_{L^2}.$$

Moreover, these estimates are sharp up to factors $N^{\varepsilon}$ for arbitrarily small $\varepsilon$. This conjecture predicts the optimal generic-in-parameter long-time Strichartz bounds for Schrödinger evolution on irrational rectangular tori; the use of genericity is justified by metric Diophantine approximation, while the sharpness assertion and the full range of estimates remain open in the stated generality.