On sums of two and three roots of unity
Let \(\alpha \) be a sum of \(k\) distinct \(N\)th roots of unity, where \(2\leq k<N\), and let \(A(k,N)\) be the set of such numbers \(\alpha .\) In [Am. Math. Mon. 93, 457--459 (1986; Zbl 0602.10026)] \textit{G. Myerson}, raised the following two problems. How small can \(\left| \alpha \right| \) be? How large can the modulus of the product of all conjugates of \(\alpha \) lying in the unit disc, say \(\wedge (\alpha )\), be? In the paper under review, the author gives a ``complete'' answer to these questions for \(k=2,\) by proving the theorem below. Let \(\alpha :=e^{i2\pi a/N}+e^{i2\pi b/N}\), where \(a\) and \(b\) are rational integers satisfying \(0\leq a<b\leq N-1\), and let \(n:=N/\gcd (N,b-a)\). Then, the maximum and the minimum of the absolute values of the conjugates of \(\alpha \) are, respectively, \(\sqrt{2(1+\cos (2\pi /n))}\) and \(\sqrt{2(1+\cos (2\pi j/n))}\), where \(j\) is the greatest positive rational integer satisfying \(\gcd (j,n)=1\) and \(j\leq n/2\); thus \(\alpha \) has no conjugate lying in unit disc for \(n\in \{3,4,6,10\}\), and \(\alpha =0\) when \(n=2\). Moreover, if \(n\notin \{2,3,4,6,10\}\), then \[ 0<\wedge (\alpha )^{1/\deg (\alpha )}=\prod\limits_{\overset{n/3<j<2n/3}{\gcd (j,n)=1}}\left| 2\cos (\pi j/n)\right| ^{1/\varphi (n)}<1, \] where \(\varphi \) is the Euler phi function, and so \(\lim_{n\rightarrow \infty }\wedge (\alpha )^{1/\deg (\alpha )}=0.723926111\dots\), which is the inverse of the Mahler measure of the polynomial \(1+x+y\). Also, he shows that there is an effectively computable constant \(\rho <1\) such that \(\wedge (\alpha )^{1/\deg (\alpha )}\leq \rho \) for all \(\alpha \in A(3,N)\), when \(N\) is a sufficiently large prime. Finally, the author conjectures that the set \(\{\lim_{N\rightarrow \infty}\wedge (\alpha )^{1/\deg (\alpha )}\mid \alpha \in A(k,N)\) and \(N\) is prime\(\}\), where \(k\) is fixed, has a unique limit point, and exhibits some related results.
scientific article; zbMATH DE number 6943088