Low regularity analysis of the Zakharov--Kuznetsov equation on $\mathbb{R} \times \mathbb{T}$

We consider the Cauchy problem for the Zakharov-Kuznetsov equation in the cylinder. We improve the local wellposedness to spaces of regularity $s > 1/2$. The result is optimal in terms of the corresponding bilinear estimate or Picard iteration. Our method is based on an improvement of the understanding of the resonant set, identifying and exploiting its particular geometric properties. We also consider the problem under randomization of the initial data, in which case we obtain solutions for generic data in $H^{s}$ for some $s < 0$. To do so, we consider a novel approach based on lower regularity modifications of the classical $X^{s, b}$ spaces that allow to control concentration of mass in small sets of frequencies.