On Sparse Holes and a Finite-Fold Problem of Nathanson Concerning Minimal Additive Complements

Let W ⊆ Z W⊆ Z upper W subset of or equal to double struck upper Z be bounded below and normalised by inf W = 1 inf W=1 inf upper W equals 1 , and put W ― = Z > 0 ∖ W W= Z>0 W upper W overbar equals double struck upper Z Subscript greater than 0 Baseline minus upper W . Nathanson asked whether infinite sets of integers admit minimal additive complements. Chen and Yang [‘On a problem of Nathanson related to minimal additive complements’, SIAM J. Discrete Math. 26 (2012), 1532–1536] proved that the two-sided case is positive and that sufficiently large consecutive gaps in W ― W upper W overbar force nonexistence in the one-sided case. Chen and Ding [‘On a problem of Nathanson on nonminimal additive complements’, Bull. Aust. Math. Soc. 114 , 6–13] isolated a sparse-hole condition under which no ordinary minimal complement exists. We prove a finite-fold version of this sparse-hole obstruction. For an integer h ≥ 1 h≥ 1 h greater than or equals 1 , write h C = C + ⋯ + C hC=C+⋯ +C h upper C equals upper C plus midline horizontal ellipsis plus upper C . We show that if W ― W upper W overbar<