Let X be a finite set of complex numbers and let A be a normal operator with spectrum X 
   that acts on a separable Hilbert space H. Relative to a fixed orthonormal basis 
   e_1,e_2, ... for H, A gives rise to a matrix whose diagonal is a sequence d=(d_1,d_2,...) 
   with the property that each of its terms d_n belongs to the convex hull of X. 
   Not all sequences with that property can arise as the diagonal of a normal operator 
   with spectrum X.
   
    The case where X is a set of real numbers has received a great deal of attention 
    over the years, and is reasonably well (though incompletely) understood. In this 
    paper we take up the case in which X is the set of vertices of a convex polygon 
    in the complex plane. The critical sequences d turn out to be those that accumulate 
    rapidly in X in the sense that $$ \sum_{n=1}^\infty {\rm{dist}} (d_n,X)<\infty. $$
    
    We show that there is an abelian group $\Gamma_X$ -- a quotient of $R^2$ by a 
    countable subgroup with concrete arithmetic properties -- and a surjective mapping 
    of such sequences $d\mapsto s(d)\in\Gamma_X$ with the following property: If s(d) is not 0, 
    then d is not the diagonal of any such operator A.
    
    We also show that while this is the only obstruction when X contains two points, 
    there are other (as yet unknown) obstructions when X contains more than two points.