  Poissonian pair correlations have sparked interest within the mathematical
community, because of their number theoretic properties, and their connections
to quantum physics and probability theory, particularly uniformly distributed
random numbers. Rather recently, several generalizations of the concept have
been introduced, including weak Poissonian pair correlations and $k$-th order
Poissonian correlations. In this paper, we propose a new generalized concept
called $(k,m,\beta)$-Poissonian box correlations. We study their properties and
more specifically their relation to uniform distribution theory, discrepancy
theory, random numbers and gap distributions.
