On the Peres--Schlag orthogonal projection problem and Kakeya-type sets

We investigate the Peres–Schlag nonempty interior problem for orthogonal projections in both the finite-field and Euclidean settings. Over finite fields ${\mathbb{F}}_q^n$, we employ the polynomial method to establish sharp projection results, and uncover a new connection with stability versions of the finite-field $(n,m)$-set problem. Over Euclidean spaces ${\mathbb{R}}^n$, we obtain improved nonempty interior results beyond those of Peres and Schlag in certain parameter ranges. Our proof combines techniques from geometric measure theory and harmonic analysis, including $L^p$-estimates for Kakeya maximal operators and maximal $k$-plane transforms.