Solution to a problem of Bombieri (Q1327580)

The author proves the following result: If \(b_ 1, b_ 2, \dots\) is a sequence of real numbers, satisfying (as \(m \to \infty)\) (1) \(| b_ m | \leq 1 + o(1)\), and, for each \(m > 1\), (2) \(m b_ m \sum^{m - 1}_{k = 1} b_ k b_{m - k} + o(m)\), then one of the following cases holds for every \(m \geq 1\): \(b_ m = o(1)\) or \(b_ m = (-1)^ m + o(1)\) or \(b_ m = 1 + o(1)\). This result contains [more than] a solution of a problem posed by \textit{E. Bombieri} [see ibid. VIII. Ser. 36, 252-257 (1963; Zbl 0121.282) and ibid IX. Ser. 1, No. 3, 177-179 (1990; Zbl 0714.11060)]: If \(a_ 1, a_ 2, \dots\) is a sequence of non-negative real numbers satisfying (2') \(ma_ m + \sum^{m - 1}_{k = 1} a_ k a_{m-k} = 2m + O(1)\) [in fact, the weaker error term \(o(m)\) is sufficient!], then \(a_ m = 1 + o(1)\) or \(a_ m = 1 - (-1)^ m + o(1)\).