The author presents a new approach to the study of combinatorial structures via group ring equations.
In a fundamental paper of R. J. Turyn [Character sums and difference sets, Pac. J. Math. 15, 319-346 (1965; Zbl 0135.05403)] it is shown that the character method for the study of group ring equations works nicely under the so-called self-conjugacy condition. Recall that an integer is called self-conjugate modulo if all prime ideals above in the th cyclotomic field (with ) are invariant under complex conjugation. Under this condition all cyclotomic integers in of absolute value can be determined for any integer . The complete knowledge of the cyclotomic integers of prescribed absolute value is the key ingredient making the character method work so well under the self-conjugacy condition. However, self-conjugacy is a very severe restriction, i.e. the self-conjugacy method fails in almost all cases, because the ‘probability’ that is self-conjugate modulo decreases exponentially fast in the number of distinct prime divisors of and . Knowing the cyclotomic integers of prescribed absolute value completely would lead to an almost complete determination of the class group of the underlying cyclotomic field modulo the class group of its maximal real subfield. This, however, is a problem of algebraic number theory that appears to be far beyond the scope of the methods known today. This demonstrates the needs for more general results about cyclotomic integers of prescribed absolute value.
The author presents a new approach to the absolute value problem. Exploiting the decomposition groups of prime ideals the following key result is proved, where is an integral valued function the definition of which is too involved to be restated here. Theorem 3.5. Assume for where and are positive integers. Then for some .
This reduction to subfields is the key to obtain a general bound on the absolute value of cyclotomic integers, the upshot of which is Theorem 4.2. Let be of the form where are integers with for some constant . Furthermore, assume that is an integer. Then where is the number of distinct odd prime divisors of . If the assumption on the coefficients is replaced by , then where is the number of distinct prime divisors of .
The preceeding theorems are applied to derive a new general exponent bound for difference sets in groups: Theorem 5.2. Assume the existence of a -difference set in a group . If is a normal subgroup of such that is cyclic of order , then where is the number of distinct odd prime divisors of . This result is then applied to all parameter series corresponding to known difference sets with , i.e. to Hadamard, McFarland, Spence and Chen/Davis/Jedwab parameters. The results obtained here have strong implications for the (non-)existence of circulant Hadamard matrices, which is outlined at the end of Section 6. Section 7 presents a general exponent bound for groups containing relative difference sets. With this at hand, the author derives strong necessary conditions for the existence of quasiregular projective planes which in turn lead to asymptotic exponent bounds for abelian groups admitting planar functions. Finally, in Section 8, the methods developed in the paper are utilized for the study of group invariant weighing matrices.