DIFFERENCE SETS AND ALGEBRAIC NUMBER THEORY AN EXPOSITION OF SOME RESULTS OF SCHMIDT

Abstract

Several combinatorical problems reduce to classifying certain families of difference sets. Schmidt, in [1], obtained strong new nonexistence results for this class of problem by applying algebraic number theory: a difference set can be related to a cyclotomic integer with bounded coefficients and a prescribed absolute value, and in many cases these integers can be number-theoretically proven not to exist. We lay out the core of Schmidt’s results, simplifying what we can and omitting some of the less interesting details. 1. Difference Sets A (v,k,λ,n)-difference set in a group G of order v is a k-subset D ⊂ G such that each nonzero element of G has exactly λ representations as the difference of elements in D, with n = k − λ. For instance, a Hadamard matrix is an m × m matrix H with entries ±1 satisfying H T H = mI. If we identify the row and column indices in the matrix with elements of an order-m group G, then a G-invariant Hadamard matrix is a Hadamard matrix with Hg,h = Hfg,fh for f,g,h ∈ G. Such a matrix is described by the entries H1,g; if we let D ⊂ G consist of the elements g ∈ G with H1,g = 1, then D is a difference set with v = m; with some work it turns out that n = m/4 if m> 2. In particular, we may consider circulant Hadamard matrices—Hadamard matrices invariant under cyclic groups Z/mZ. Such matrices are known only for m = 1, 4; it is known that any others must be of the form 4u 2 for some u. A conjecture holds that there exist no circulant Hadamard matrices of order greater than 4. The results described herein will rule out almost all u as candidates. To study difference sets, we first describe a few preliminaries. We identify any set S ⊂ G with the element ∑ g∈S g of the integral group ring Z[G], and for any element A = ∑ g∈G agg of the ring write A (−1) for ∑ g∈G agg−1. A character of a finite abelian group G is a homomorphism χ: G → C ∗ to the multiplicative group of nonzero complex numbers, hence to the group of e-th roots of unity for some e called the order of χ. We extend group homomorphisms G → H linearly to ring homomorphisms Z[G] → Z[H]

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