Journal of number theory vol:117 issue:1 pages:14-30
For an expanding matrix H epsilon Z(kxk), a subset W subset of Z(k) is called a complete digit set, if all points of the integer lattice Z(k) can be uniquely represented as a finite sum x = Sigma(N(x))(i=0) H(i)r(i), with r(i) epsilon W and N(x) epsilon N. We present a necessary and sufficient condition for the existence of a complete digit set in case vertical bar det(H)vertical bar = 2, implying that W is a binary complete digit set. This allows a characterization of the binary number systems (H, W) in Z(k). It is shown that, when H has a complete digit set, all its complete digit sets form a finitely generated Abelian group. Complete lists are given for dimension k = 1 to 6. (c) 2005 Elsevier Inc. All rights reserved.