I wonder if it is possible to show, without using the Schmidt subspace/Roth theorem/Baker's bounds on linear forms in logarithms or other very deep results, that, in a number field, not all integral elements are sums of two units/most of them are not.

The known results that state that there are few elements that are sums of two units in number fields (see e.g. Fuchs, Tichy, Ziegler, "On quantitative aspects of the unit sum number problem", Arch. Math. 93 (2009)) seem to use either Thue-Siegel-Roth theorem or Baker's bounds on linear forms in logarithms. Both I find very difficult to reach. Since I rely on a similar result in a statement I have made [1], [2], [3], the motivation behind the question is a wish to find a simpler argument (or to convince oneself that existence of (many) elements that are not sums of two units is a claim that is itself of strength that is seemingly out of reach to significantly easier means than Thue-Siegel-Roth/Baker theorems).

References:

[1] Zinevičius A., *On the congruent number problem over integers of real number fields*, Albanian J. Math. **8** (2014), 49-53.

[2] Zinevičius A., *On the congruent number problem over integers of cyclic extensions*, Math. Slovaca **66** (2016), 561-564,

[3] Zinevičius A., *Corrigendum to "On the congruent number problem over integers of cyclic extensions"*, Math. Slovaca **69** (2019), 1233-1233.

somenumber field: just take the two solutions of $X^2 = \alpha X + 1$. (By the way, there is a typo in my comment above: the chapter of Bombieri and Gubler given the Pade approximations solution is the 5th one, not the 4th.) $\endgroup$1more comment