  The purpose of this note is to present a construction of sequences which do
not have metric Poissonian pair correlations (MPPC) and whose additive energies
grow at rates that come arbitrarily close to a threshold below which it is
believed that all sequences have MPPC. A similar result appears in work of
Lachmann and Technau and is proved using a totally different strategy. The main
novelty here is the simplicity of the proof, which we arrive at by modifying a
construction of Bourgain.
