Joint IAS/Princeton University Number Theory Seminar

A subset $D$ of a finite cyclic group $Z/mZ$ is called a "perfect difference set" if every nonzero element of $Z/mZ$ can be written uniquely as the difference of two elements of $D$. If such a set exists, then a simple counting argument shows that $m=n^2+n+1$ for some nonnegative integer $n$. Singer constructed examples of perfect difference sets in $Z/(n^2+n+1)Z$ whenever $n$ is a prime power, and it is an old conjecture that these are the only such n for which a perfect difference set exists. In this talk, I will discuss a proof of an asymptotic version of this conjecture: the number of $n$ less than $N$ for which $Z/(n^2+n+1)Z$ contains a perfect difference set is $\sim N/\log(N)$.