By Lisa Jacobsen
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Additional info for Analytic Theory of Continued Fractions III: Proceedings of a Seminar Workshop
Example text
1. Parameter spaces for periodic sets. We want to study the more general situation of periodic sphere packings in greater detail. 18) ti + L Λ = i=1 in Rd is given by a lattice L ⊂ Rd , together with m translation vectors ti ∈ Rd , i = 1, . . , m. d for lattices. 19) ti + Z d . Λt = i=1 Here, A ∈ GLd (R) satisfies in particular L = AZd . Since we are only interested d , in properties of periodic sets up to isometries, we encode Λ by Q = At A ∈ S>0 together with the m translation vectors t1 , . .
22) The condition λ(X) ≥ λ gives infinitely many linear inequalities pv (X) = Q[v] = X, (vv t , 0) ≥ λ for v ∈ Zd \{0}, as in the case m = 1. 23) where i, j ∈ {1, . . , m} with i = j and v ∈ Zd . These polynomials are of degree 3 in the parameters qkl , tkl of X. Note that they are linear for a fixed t. Observe also that pi,m,v and pm,j,v are special due to our assumption tm = 0 and that there is a symmetry pi,j,v = pj,i,−v by which we may restrict our attention to polynomials with i ≤ j. In case of equality i = j we have the linear function pi,i,v = pv .
Lagrange’s reduction domain and a reduced basis. 2 . Thinking of lattices in R2 , a base A = (a1 , a2 ) ∈ is a reduction domain in S>0 GL2 (R) is reduced, if 0 ≤ at1 a2 ≤ 12 a1 2 and a1 ≤ a2 . The first condition implies that the angle between a1 and a2 is between 60 and 90 degrees. By the second condition we see that a1 is the shortest non-zero vector of the lattice. In Figure 1, D is shown as a three dimensional polyhedral cone, together with possible positions of a vector a2 in a reduced basis (a1 , a2 ).
