Zeros of the Brownian Sheet

In this work, we firstly answer to a question raised by Khoshnevisan (Trans Am Math Soc 359:3125-3151, 2007) [Open Problem 4] by proving that almost surely there is no projection of big enough rank changing the Hausdorff dimension of the zeros of the Brownian sheet. Secondly, we prove that almost surely for every projection whose rank is not matching the aforementioned condition, the projection of the zero set is the entirety of the projective space.