  $k$-level correlation is a local statistic of sequences modulo 1, describing
the local spacings of $k$-tuples of elements. For $k = 2$ this is also known as
pair correlation. We show that there exists a well spaced increasing sequence
of reals with additive energy of order $N^3$ and Poissonian $k$-level
correlation for all integers $k \geq 2$, answering in the affirmative a
question raised by Aistleitner, El-Baz, and Munsch. The construction is
probabilistic, and so we do not obtain a specific sequence satisfying this
condition. To prove this, we show that random perturbations of a sequence with
small discrepancy gives, almost surely, a sequence with Poissonian $k$-level
correlation, a fact which may be of independent interest.
