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We characterize those derivations from the convolution algebra ℓ 1 (ℤ + ) to its dual that are weakly compact, providing explicit examples that are not compact. The characterization is combinatorial, in terms of ‘translation‐finite’ subsets of ℤ + , and we investigate how this notion relates to other notions of ‘smallness’ for infinite subsets of ℤ + . In particular, we prove that a set of strictly positive Banach density cannot be translation‐finite; the proof has a Ramsey‐theoretic flavour.

Translation‐finite sets and weakly compact derivations from ℓ 1 (ℤ + ) to its dual