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.
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