Sparse norm preservation from entrywise Gram control
= Sparse norm preservation from entrywise Gram control
If $E=A^TA/d-I$ has <entrywise maximum norm> at most $t$, every vector $z$ supported on at most $s$ coordinates satisfies $|z^TEz|\leq t\|z\|_1^2\leq st\|z\|_2^2$. Thus the squared norm distortion is at most $st$ on all such sparse vectors. Use the quadratic inequality at $z=0$, where a ratio would be undefined.