An additive complement of a set 
of nonnegative integers
is a set 
of nonnegative integers
such that the sumset 
contains all sufficiently large integers.
For infinite additive complements, write 
and 
for the numbers of their elements at most 
.
Zhang (2026) gave a discrete proof of the sharp implication
 |
The threshold is strict. At equality, examples have 
for infinitely many 
.
Van Doorn et al. (2026) independently proved the sharp 
obstruction. For infinite additive complements satisfying
, their result gives
 |
Van Doorn et al. (2026) attribute an initial proof to AI and report a Lean formalization checked by the authors.
