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

Let $\alpha>0$ be fixed. For a matrix $A$ as above, let $J_p^\alpha(A) = \{j=1,...,p: \|A e_j\|_1 \ge p^\alpha\}$ be the indices of the columns with "large" $\ell_1$ norm.

• How fast does $$u_p^\alpha = \sup_{A\in \mathcal S(p)} |J_p^\alpha(A)|$$ 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 this bound for $A$ with non-negative entries $p^\alpha |J| \le \sum_{j\in J} \|c_j\|_1\le \sum_{ij}|a_{ij}|=\sum_{ij} a_{ij} \le p \|A\|_{op}$, but I suspect that the general case will be much different.