The following is known:
There exists a positive constant $c>0$ such that for every prime $p$ and every subset $A$ of $\Bbb Z_p$ of size $k$, there exists an element $x \in \Bbb Z_p$ such that the set $xA$ intersects every interval in $\Bbb Z_p$ of length at least $\frac{cp}{\sqrt k}$.
I am asked to conclude from this the following:
There is a constant $d>0$ so that for every prime $p \equiv 3 \pmod 4$, every interval of length at least $d\sqrt p$ contains a quadratic residue.
I wonder What $A$ to choose, since to apply the known theorem directly I need a set that contains quadratic residues no matter by what we multiply all elements of the set, and no nonzero such set exists. We lookfor a very large subset (of size $p$ up to a constant) by the required length bound.