Kronecker Constants for Finite Subsets of Integers

A set of integers S is called ε-Kronecker if every function on S of modulus one can be approximated uniformly to within ε by a character. The least such ε is called the ε-Kronecker constant. We transform the problem of calculating ε-Kronecker constants for finite sets of d elements into a geometric optimization problem. Using this approach we can explicitly determine the ε-Kronecker constant for any two element set and deduce a (non-trivial) upper bound for any finite set. Kronecker constants are determined for many classes of three element sets, including all sum sets, product sets and arithmetic progressions. The answers are surprisingly complicated.