An asymptotic version of the prime power conjecture for perfect difference sets

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=n2+n+1 for some nonnegative integer n. Singer constructed examples of perfect difference sets in Z/(n2+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/(n2+n+1)Z contains a perfect difference set is ∼N/log(N).