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

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

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Sarah Peluse, Institute for Advanced Study

Zoom link:  https://princeton.zoom.us/j/97126136441

Password: the three digit integer that is the cube of the sum of its digits

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 ~N/log(N).