We recall the notion of nearest integer continued fractions over the
Euclidean imaginary quadratic fields K and characterize the "badly
approximable" numbers, (z such that there is a C(z)>0 with β£zβp/qβ£β₯C/β£qβ£2 for all p/qβK), by boundedness of the partial quotients in the
continued fraction expansion of z. Applying this algorithm to "tagged"
indefinite integral binary Hermitian forms demonstrates the existence of entire
circles in C whose points are badly approximable over K, with
effective constants.
By other methods (the Dani correspondence), we prove the existence of circles
of badly approximable numbers over any imaginary quadratic field, with loss of
effectivity. Among these badly approximable numbers are algebraic numbers of
every even degree over Q, which we characterize. All of the examples
we consider are associated with cocompact Fuchsian subgroups of the Bianchi
groups SL2β(O), where O is the ring of integers in an
imaginary quadratic field.Comment: v3: Improved exposition (hopefully), especially in the second half of
the pape