Abstract.
It is shown that if R is a finitely generated integral domain of zero characteristic, then for every n there exist elements of R which are not sums of at most n units. This applies in particular to rings of integers in finite extensions of the rationals. On the other hand there are many infinite algebraic extensions of the rationals in which every integer is a sum of two units.
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Jarden, M., Narkiewicz, W. On Sums of Units. Mh Math 150, 327–332 (2007). https://doi.org/10.1007/s00605-006-0402-z
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DOI: https://doi.org/10.1007/s00605-006-0402-z