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On the critical determinants of certain star bodies

Werner Georg Nowak (2017)

Communications in Mathematics

In a classic paper, W.G. Spohn established the to-date sharpest estimates from below for the simultaneous Diophantine approximation constants for three and more real numbers. As a by-result of his method which used Blichfeldt’s Theorem and the calculus of variations, he derived a bound for the critical determinant of the star body | x 1 | ( | x 1 | 3 + | x 2 | 3 + | x 3 | 3 ) ≤ 1 . In this little note, after a brief exposition of the basics of the geometry of numbers and its significance for Diophantine approximation, this latter result is improved...

On the Euclidean minimum of some real number fields

Eva Bayer-Fluckiger, Gabriele Nebe (2005)

Journal de Théorie des Nombres de Bordeaux

General methods from [3] are applied to give good upper bounds on the Euclidean minimum of real quadratic fields and totally real cyclotomic fields of prime power discriminant.

On the magnitudes of some small cyclotomic integers

Frederick Robinson, Michael Wurtz (2013)

Acta Arithmetica

We prove the last of five outstanding conjectures made by R. M. Robinson from 1965 concerning small cyclotomic integers. In particular, given any cyclotomic integer β all of whose conjugates have absolute value at most 5, we prove that the largest such conjugate has absolute value of one of four explicit types given by two infinite classes and two exceptional cases. We also extend this result by showing that with the addition of one form, the conjecture is true for β with magnitudes up to 5 + 1/25....

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