  We study a fundamental measure for wireless interference in the SINR model
known as (weighted) inductive independence. This measure characterizes the
effectiveness of using oblivious power --- when the power used by a transmitter
only depends on the distance to the receiver --- as a mechanism for improving
wireless capacity.
  We prove optimal bounds for inductive independence, implying a number of
algorithmic applications. An algorithm is provided that achieves --- due to
existing lower bounds --- capacity that is asymptotically best possible using
oblivious power assignments. Improved approximation algorithms are provided for
a number of problems for oblivious power and for power control, including
distributed scheduling, connectivity, secondary spectrum auctions, and dynamic
packet scheduling.
