On the minimum of the unit lattice. (English) Zbl 0755.11034

Let \(K\) be a finite extension of \(\mathbb{Q}\) of degree \(n\) with maximal order \(R\). As usual, let \(r_ 1\) and \(2r_ 2\) be the number of real and non- real embeddings of \(K\) into \(\mathbb{C}\), and let \(r=r_ 1+r_ 2-1\) be the rank of the unit group \(R^*\). Let Log denote the standard mapping of \(K^*\) into \(\mathbb{R}^ r\) so that \(L:=\text{Log}(R^*)\) is a lattice of rank \(r\). Let \(\lambda(L)\) denote the minimum of the Euclidean norms of the non-zero vectors in \(L\). Using results of C. J. Smyth [Bull. Lond. Math. Soc. 3, 169-175 (1971; Zbl 0235.12003)], and E. Dobrowolski [Acta Arith. 34, 391-401 (1979; Zbl 0416.12001)], the author proves the following lower bound: \[ \lambda(L)>{1\over\sqrt{r+1}}{1\over1000}\left({\log\log n\over\log n}\right)^ 3. \]


11R27 Units and factorization
11H06 Lattices and convex bodies (number-theoretic aspects)
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