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We now turn to several specific examples of this principle in various contexts.  We begin with the more “combinatorial” or “non-ergodic theoretical” instances of this principle, in which there is no underlying probability measure involved; these situations are simpler than the ergodic-theoretic ones, but already illustrate many of the key features of this principle in action.

An alternate argument for the arithmetic large sieve inequality September 2008 This short note describes a very natural and well-motivated derivation of the "arithmetic" large sieve inequality from the dual of the analytic inequality, which avoids the usual trick of submultiplicativity of Gallagher. This is also described in a blog post.

This earlier problem is of considerable importance in algorithmic number theory, and both have been excellent testing and breeding grounds for various important techniques, notably (and this is close to my heart…) leading to the invention and development of the first “large sieve” method by Linnik.