Character sums and difference sets

This paper concerns difference sets in finite groups. The approach is as follows: if D is a difference set in a group G, and any character of G, (D) ->x(7) is an algebraic integer of absolute value V~n in the field of mth roots of 1, where m is the order of . Known facts about such integers and the relations which the (D) must satisfy (as varies) may yield information about D by the Fourier inversion formula. In particular, if (D) is necessarily divisible by a relatively large integer, the number of elements g of D for which (g) takes on any given value must be large; this yields some nonexistence theorems.