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Let $K$ be a number field. Let $\mathcal{O}_K^{*}$ be the units in the ring of integers of this field.

I am interested in knowing how many units $u,v \in \mathcal{O}_K^{*}$ exist such that $u + v$ is also a unit. This is equivalent to knowing units $u$ such that $1+u$ is also a unit.

In particular, I'm interested when we have cyclotomic number fields as $K$. Some easy examples can be constructed here by using elements of the form $1+\zeta_n^m$.

Are there any results of this type, perhaps about density of such units $u$? Where is the starting point for this?

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    $\begingroup$ A unit $u$ is called exceptional if $1-u$ is also a unit, so that gives you a search term. There is a fair bit of literature about them. Lenstra studied them, he might be responsible for the terminology. $\endgroup$ Sep 24 at 22:40
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    $\begingroup$ I believe that results on the $S$-unit equation will also be relevant. $\endgroup$
    – Will Sawin
    Sep 24 at 23:56
  • $\begingroup$ Please use a high-level tag like "nt.number-theory". I added this tag now. $\endgroup$
    – GH from MO
    Sep 25 at 2:32
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    $\begingroup$ You might look at some of Morris Newman's papers, such as 'Consecutive units' in Proc. Amer. Math. Soc. 108 (1990), no. 2, 303–306. $\endgroup$
    – Stopple
    Sep 25 at 17:17

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I only want to add to Gerry's comment that it is a well-known result by Siegel that the number of exceptional units for a fixed number field is finite. There are even bounds on this number by Evertse, which only depend on the degree of the field extension. There also exist examples of number fields (by Triantafillou) where there does not exist a single exceptional unit. (This would have been a comment, but I am currently not able to make them)

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