On the distribution of balanced subgroups

Let n>2 be an integer. Any element of (Z/nZ)× has a unique representative in one of the two intervals (0,n/2) and (n/2,n). A subgroup H of (Z/nZ)× is balanced if every coset of H intersects these two intervals equally. Such subgroups have repercussions to ranks of elliptic curves over function fields. For a fixed integer g≠0,1, how frequently is the cyclic subgroup 〈gmodn〉 balanced? We prove a conjecture of Pomerance and Ulmer that this distribution is essentially determined by two special families of balanced subgroups. One of the deeper tools that we use is a theorem of Grantham which counts primes in a residue class that split completely in a given number field.