Optimal support cardinality conjecture for the discrete Schrödinger equation

Let $B_d(N)$ be the $d$-dimensional box of radius $N$, and let $S_d(N)$ denote the minimum support cardinality of a nonzero solution of the discrete Schrödinger equation on $B_d(N)$ that satisfies $u(\mathbf 0)\ne 0$. Here, $\Omega$ denotes a lower bound up to a positive constant, and $\lceil d/2\rceil$ is the least integer greater than or equal to $d/2$.

Support cardinality conjecture. For $d\geq 1$, we have

$$S_d(N)=\Omega\left(N^{\lceil d/2\rceil}\right).$$

The conjecture asserts that the exponent supplied by the explicit sparse constructions is optimal in every dimension. It is consistent with the exponent-sharp bounds established in dimensions $1$, $2$, and $3$, and with the resulting exponent-sharp estimate in dimension $4$ up to logarithmic factors; the corresponding sharp lower bounds in higher dimensions remain unknown.