Peres--Schlag's nonempty-interior problem and a shifted-product variant for product sets

We study finite-field analogues of the Peres–Schlag nonempty-interior problem for product sets. Given $A\subseteq {\mathbb{F}}_p$, we ask when a suitable one-dimensional linear image of $A^n$ is full; equivalently, when there exist coefficients $t_1,\dots ,t_n\in {\mathbb{F}}_p$ such that $$t_{1}A+\cdots +t_{n}A={\mathbb{F}}_p\mathrm{.}$$ For $n\ge 3$, we prove that, for every $\eta >0$, this holds whenever $$\mathrm{\mid }A\mathrm{\mid }{\gg }_{n,\eta }{p}^{\frac{3}{2n-1}+\eta }\mathrm{.}$$ This improves the exponent predicted by the direct product-set analogue of the Peres–Schlag threshold, namely $\mathrm{\mid }A\mathrm{\mid }\gg p^{2\mathrm{/}n}$. We also prove a two-dimensional near-half-density result. Motivated by sum-product phenomena, we also introduce and study a product-type variant in which linear forms are replaced by shifted product maps. We prove finite-field covering results for shifted products $$(t_1+A)(t_2+A)\cdots (t_n+A)$$ at the same density scale as in the linear case. Finally, we prove a Euclidean shifted-product analogue: if $A\subseteq \mathbb{R}$ is Borel and ${\dim }_{H}A>2\mathrm{/}n$, then some shifted product of $n$ copies of $A$ contains a nonempty open interval.