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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, 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).

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.

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, I wish to find a simpler argument.

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, 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).

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.

Thank you.

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, I wish to find a simpler argument.