Let Z+ and Z− denote
the sets of positive and negative integers respectively. We study relations
between various thinness conditions on subsets E of Z+, with particular
emphasis on those conditions that imply Z−∪ E is a set of continuity. For
instance, if E is a Λ(1) set, a p-Sidon set (for some p < 2), or a UC-set,
then E cannot contain parallelepipeds of arbitrarily large dimension, and it
then follows that Z−∪ E is a set of continuity; on the other hand there is a
set E that is Rosenthal, strong Riesz, and Rajchman, which is not a set of
continuity.