  In this paper we consider the fractional parts of a general sequence, for
example the sequence $\alpha \sqrt{n}$ or $\alpha n^2$. We give a general
method, which allows one to show that long-range correlations (correlations
where the support of the test function grows as we consider more points) are
Poissonian. We show that these statements about convergence can be reduced to
bounds on associated Weyl sums. In particular we apply this methodology to the
aforementioned examples. In so doing, we recover a recent result of
Technau-Walker (2020) for the triple correlation of $\alpha n^2$ and generalize
the result to higher moments. For both of the aforementioned sequences this is
one of the only results which indicates the pseudo-random nature of the higher
level ($m \ge 3$) correlations.
