On the magnitudes of some small cyclotomic integers (Q2851131)

For a cyclotomic integer \(\beta\), let the \textit{house} of \(\beta\), denoted \(|\overline{\beta}|\) be the largest absolute value of all conjugates of \(\beta\). In [Math. Comput. 19, 210--217 (1965; Zbl 0129.02303)], \textit{R. M. Robinson} made a study of small cyclotomic integers \(\alpha\), all of whose conjugates lie in \(|z|\leq R\) for \(R=2\) and \(R=\sqrt{5}\). Robinson made five conjectures concerning these numbers, four of which were proved by \textit{A. Schinzel} [Acta Arith. 11, 419--432 (1966; Zbl 0151.03901)], \textit{J. W. S. Cassels} [J. Reine Angew. Math. 238, 112--131 (1969; Zbl 0179.35203)] and \textit{A. J. Jones} [Proc. Camb. Philos. Soc. 64, 673--682 (1968; Zbl 0162.06804); ibid. 66, 43--59 (1969; Zbl 0175.04403)]. In this paper, the authors resolve the final outstanding of these conjectures: Robinson conjecture 4: If \(\beta\) is a cyclotomic integer with \((|\overline{\beta}|)^2\leq 5\) then \(|\overline{\beta}|\) has one of the two forms \(2\cos(\pi/N)\), \(\sqrt{1+4\cos^2(\pi/N)}\), where \(N\) is a positive integer, or else is equal to one of the two numbers \(\sqrt{\frac{5+\sqrt{13}}{2}},\;\frac{\sqrt{7}+\sqrt{3}}{2}\).NEWLINENEWLINEThese values do actually occur for some cyclotomic integers (with the exception of \(N=1\)), specifically for \(\beta\) of form \(\zeta_N+\zeta_N^{-1},\;\zeta_4+\zeta_N+\zeta_N^{-1},\;1+\zeta_{13}+\zeta_{13}^{4}\) and \(\zeta_{84}^{-9}+\zeta_{84}^{-7}+\zeta_{84}^3+\zeta_{84}^{27}\). The authors follow the approach of Cassels [loc. cit.] and also of Calegari, Morrison and Snyder [\textit{F. Calegari} et al., Commun. Math. Phys. 303, No. 3, 845--896 (2011; Zbl 1220.18004)] for totally real \(\beta\). The authors improve also this result by the following: Theorem. If \(\beta\) is a cyclotomic integer with \((|\overline{\beta}|)^2\leq 5+\frac{1}{25}\) then, either \(\beta\) is a number of the list above, or \(|\overline{\beta}|=|1+\zeta_{70}+\zeta_{70}^{10}+\zeta_{70}^{29}|\), where \(\zeta_{70}=e^{\frac{2\pi i}{70}}\).NEWLINENEWLINESee also \textit{J. H. Loxton} [Acta Arith. 22, 69--85 (1972; Zbl 0217.04203)] for a qualitative method, but with non effective arguments.