Additive Groups with Many Unimodular Elements

In this note, we prove that for every even integer d ≥ 4, there is a number field K of degree d over Q, such that for each sufficiently large N ∈ N among (2N + 1)d distinct elements of the form a1ω1 + · · · + adωd , where ω1, . . . , ωd is an integral basis of the ring OK and a1, . . . , ad ∈ Z, |a1|, . . . , |ad| ≤ N, at least c(log N)d/2−1 elements are unimodular. Here, c is a positive constant that depends on the field and the integral basis only. In particular, for a quartic field K, the lower bound for the number of unimodular elements of such form is at least c log N. We show that that this estimate is best possible for any quartic field K and any ω1, . . . , ω4 ∈ OK .