  In this article we study the possible size of support of solutions to the
discrete stationary Schrodinger equation $\Delta u(x)+V(x)u(x)=0$ in
$\mathbb{Z}^d$. We show that for any nonzero solution to any discrete
stationary Schrodinger equation the dimension of the support is at least
$\log_2(d)-7.$
  In the related setting of $\mathbb{Z}_2$-valued harmonic functions in
$\mathbb{Z}^d$ one can improve the estimate on support's dimension to
$\log_2(d).$
  However, we also provide an example where a $\mathbb{Z}_2$-valued harmonic
function in $\mathbb{Z}^d$ has a fractal-like support with dimension
$\log_2(d)+1$. This fractal satisfies a recurrence relation: $$X =
2X+\{e_1,-e_1,\dots,e_d,-e_d\}.$$
  This example and estimate provide an answer to the Malinnikova's question
about the smallest size of set $X\subset\mathbb{Z}^d$ such that no cross
contains exactly one point of $X$.
