Consider the set $\mathcal S(p)$ of symmetric matrices $A$ of size $p\times p$ with bounded spectrum, say, $\lVert A\rVert_\text{op}\le 10$ and $\lVert A^{-1}\rVert_\text{op}\le 10$.

Let $\alpha>0$ be fixed. For a matrix $A$ as above, let $J_p^\alpha(A) = \{j=1,\dotsc,p: \lVert A e_j\rVert_1 \ge p^\alpha\}$ be the indices of the columns with "large" $\ell_1$ norm.

• How fast does $$u_p^\alpha = \sup_{A\in \mathcal S(p)} \lvert J_p^\alpha(A)\rvert$$ grow as $p\to\infty$? For any value of $\alpha$, both non-trivial upper and lower bounds would be interesting.
• Are there some phase transition with respect to $\alpha$, e.g., some $\alpha_*$ such that $u_p^\alpha$ grows very differently for $\alpha>\alpha_*$ and $\alpha<\alpha_*$?

Of course, one has the bound for $A$ with non-negative entries $p^\alpha \lvert J\rvert \le \sum_{j\in J} \lVert c_j\rVert_1\le \sum_{ij}\lvert a_{ij}\rvert=\sum_{ij} a_{ij} \le p \lVert A\rVert_\text{op}$, but I suspect that the general case will be much different.