ON SPARSE HOLES AND A FINITE-FOLD PROBLEM OF NATHANSON CONCERNING MINIMAL ADDITIVE COMPLEMENTS

Abstract Let W ⊆ Z $W\subseteq \mathbb Z$ 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 $\overline W=\mathbb Z_{>0}\setminus W$ 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 ― $\overline W$ 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 $h\geq 1$ h greater than or equals 1 , write h C = C + ⋯ + C $hC=C+\cdots +C$ h upper C equals upper C plus midline horizontal ellipsis plus upper C . We show that if W ― $\overline W$ upper W overbar<