Using a simple identity for the Vandermonde determinant we
recover a number of spacing results
formerly obtained by an overlapping argument of Cilleruelo
and Tenenbaum or related methods.
These include, in particular, the results on the number of
integer points on small arcs of conics of
Cilleruelo et al. and a result of Lenstra on the
number of divisors of a given integer lying in certain
arithmetical progressions.