An uncertainty principle for arithmetic sequences

Analytic number theorists usually seek to show that sequences which appear naturally in arithmetic are well-distributed'' in some appropriate sense. In various discrepancy problems, combinatorics researchers have analyzed limitations to equi-distribution, as have Fourier analysts when working with the uncertainty principle’'. In this article we find that these ideas have a natural setting in the analysis of distributions of sequences in analytic number theory, formulating a general principle, and giving several examples.