Given a set $A$ of $n$ different points in the space $(\mathbb{Z}_p)^m$ (assume $p$ is prime), and given $\delta>0$. show the following property holds for a big enough $n$ and $p$ (you can demand that $n$ is dependent on $p$; you can't demand anything from $m$ except that it's big enough so our space can actually contain $n$ different points):
Show that there exists a set of linear equations (that look like "$u\bullet v=x$") such that the number of points that don't solve any equation is in the interval $(\frac{1}{2}\pm\delta)n$. there should be $O(p \log n)$ equations.
I came across this question why studying probabilistic methods and algorithms, specifically in a chapter on the second moment method. [Please excuse the the bad translation, the questions isn't originally in English.]