Letting $\otimes$ be matrix kronecker/tensor product with $n\in\Bbb N$ as a parameter define non-negative integer vectors recursively $$v_1=\begin{bmatrix}a_1&b_1\end{bmatrix}\in\Bbb Z_{>0}^2$$ $$v_t=\begin{bmatrix}a_t&b_t\end{bmatrix}\otimes v_{t-1} \in\Bbb Z_{>0}^{2^t}$$ where $$\mbox{at every }t,t',\mbox{ }\mathsf{GCD}(a_ta_{t'},b_tb_{t'})=1\mbox{ and at every }t\neq t',\mbox{ }\mathsf{GCD}(a_tb_t,a_{t'}b_{t'})=1\mbox{ holds}$$ $$a_1,b_{1}\approx n\mbox{ and }\forall t\geq2, \mbox{ }a_t,b_{t}\approx n^{2^{t-2}}\mbox{ holds}.$$ Take non-zero integer vector $u\in\Bbb Z^{2^{t}}$ such that $\langle u,v_t\rangle=0$ at some $t\in\Bbb N$.
If $t>1$ does $L^\infty$ norm of $u$ satisfy $$n^{\frac12\big(1+\frac1{2^t-1}\big)}\leq|u|_\infty$$ with probability $1-o(1)$?
Note at $t=1$ we have $n^{}\leq|u|_\infty$.
