A Stability Result for Sparse Convolutions

Abstract: We will establish in this note a stability result for sparse convolutions on torsion-free additive (discrete) abelian groups. Sparse convolutions on torsion-free groups are free of cancellations and hence admit stability, i.e. injectivity with a universal lower bound $\alpha=\alpha(s,f)$, only depending on the cardinality $s$ and $f$ of the supports of both input sequences. More precisely, we show that $\alpha$ depends only on $s$ and $f$ and not on the ambient dimension. This statement follows from a reduction argument which involves a compression into a small set preserving the additive structure of the supports.