On sums of roots of unity
Let \(a_0,\dots,a_{k-1}\) be non-zero complex numbers. The authors consider relations \[ a_0+a_1\zeta_1+,\dots,+a_{k-1}\zeta_{k-1}=0\tag{*} \] where \(\zeta_1,\ldots,\zeta_{k-1}\) are roots of unity. Assume that (*) is decomposable, that is, no proper subsum of the left-hand side of (*) vanishes. Let \(Q\) be the smallest positive integer such that \(\zeta_i^Q=1\) for \(i=1,\dots,k-1\). In the special case that \(a_0,\dots,a_{k-1}\) are rational numbers, \textit{H. B. Mann} [Mathematika 12, 107-117 (1965; Zbl 0138.03102)] showed that \(Q\) is square free and thatall primes dividing \(Q\) are \(\leq k\); this was improved by \textit{J. H. Conway} and \textit{A. J. Jones} [Acta Arith. 30, 229-240 (1976; Zbl 0349.10014)] to \(\sum_{p|Q} (p-2)\leq k-2\) where the sum is over all primes dividing \(Q\). The authors deal with the situation that \(a_0,\dots,a_{k-1}\) belong to an algebraic number field \(L\). Let \(L_0=L\cap {\mathbb Q}(\zeta_1\cdots\zeta_{k-1})\) and \(d=[L:{\mathbb Q}]\). It follows from previous work of Loxton, Schinzel and Zannier, that \(Q\) is bounded from above by an effective bound depending only on \(k\) and \(d\). In the present paper, the authors prove the following generalisation of the results of Mann and Conway and Jones. If \(p\) is a prime and \(a\) a positive integer such that \(p^{a+1}|Q\) then \(p^a|2d\). Further, one has \[ 1+\sum_{p||Q}\Big( {p-1\over (d,p-1)}-1\Big)\leq \text{ rank}_L \{ 1,\zeta_1,\dots,\zeta_{k-1}\}\leq k-1, \] where the sum is over all primes \(p\) such that \(p|Q\) and \(p^2\nmid Q\). The proof uses linear algebra and elementary properties of cyclotomic extensions. By similar methods, the authors prove a result for characteristic functions of arithmetic progressions. Given positive integers \(q_1,\dots,q_k\) and integers \(a_1,\dots,a_k\) define functions \(\chi_i\) \((i=1,\dots,k)\) on \({\mathbb N}\) by \(\chi_i(n)=1\) if \(n\equiv a_i\) (mod \(q_i\)) and \(\chi_i(n)=0\) otherwise. Assume we have an indecomposable relation \(\sum_{i=1}^k c_i\chi_i=0\) identically on \({\mathbb N}\) with non-zero coefficients \(c_i\in{\mathbb C}\), i.e. no proper subsum of the left-hand side is identically zero. Assume gcd\((q_1,\dots,q_k)=1\) and let \(Q=\text{ lcm}(q_1,\dots,q_k)\). The authors show that if \(Q=\prod_{i=1}^r p_i^{h_i}\) with \(p_i\) prime then the \({\mathbb C}\)-vector space spanned by \(\chi_1,\dots,\chi_k\) has dimension at least \(1+\sum_{i=1}^r h_i(p_i-1)\).
scientific article; zbMATH DE number 1423216