  Using the $ku$- and $BP$-theoretic versions of Astey's cobordism obstruction
for the existence of smooth Euclidean embeddings of stably almost complex
manifolds, we prove that, for $e$ greater than or equal to $\alpha(n)$--the
number of ones in the dyadic expansion of $n$--, the ($2n+1$)-dimensional
$2^e$-torsion lens space cannot be embedded in Euclidean space of dimension
$4n-2\alpha(n)+1$. A slightly restricted version of this fact holds for
$e<\alpha(n)$. We also give an inductive construction of Euclidean embeddings
for $2^e$-torsion lens spaces. Some of our best embeddings are within one
dimension of being optimal.
