  This article deals with redundant digit expansions with an imaginary
quadratic algebraic integer with trace $\pm 1$ as base and a minimal norm
representatives digit set. For $w\geq 2$ it is shown that the width-$w$
non-adjacent form is not an optimal expansion, meaning that it does not
minimize the (Hamming-)weight among all possible expansions with the same digit
set. One main part of the proof uses tools from Diophantine analysis, namely
the theory of linear forms in logarithms and the Baker--Davenport reduction
method.
