Dynamical Systems Seminar

In the 1970s Furstenberg and Sárközy independently proved that if \(A\) is a subset of positive upper density of the integers, then \(A\) contains two distinct elements whose difference is a perfect square. Furthermore, \(A\) contains two elements whose difference is of the form \(p-1\), where \(p\) is prime. On the other hand, it is easy to see that the sets \(\{n^2 +1 \}\) and \(\{ p\}\) do not satisfy this property. What makes the sets \(\{n^2\}\) and \(\{ p-1\}\) so special, and are there other sets satisfying this property? It turns out that sets satisfying this property are none other than sets of single recurrence in ergodic theory. Are there simple ways to determine if a set satisfies this property? In this talk, I will address these questions both from a number-theoretic and ergodic-theoretic point of view, as well as related results and open problems.