MathDB
Existence of real sequence with specific properties

Source: KöMaL A. 741

February 13, 2019
algebracombinatorics

Problem Statement

Let ff be a function defined on the positive integers with f(n)≥0f(n) \ge 0 and f(n)≤f(n+1)f(n) \le f(n+1) for all nn. Prove that if ∑n=1∞f(n)n2\sum_{n = 1}^{\infty} \frac{f(n)}{n^2} diverges, there exists a sequence a1,a2,…a_1, a_2, \dots such that the sequence ann\tfrac{a_n}{n} hits every natural number, while an+m≤an+am+f(n+m)a_{n+m} \le a_n + a_m + f(n+m) holds for every pair nn, mm.
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