Additive functions in short intervals, gaps and a conjecture of Erd\H{o}s

Abstract

With the aim of treating the local behaviour of additive functions, we develop analogues of the Matom\"{a}ki-Radziwill theorem that allow us to approximate the average of a general additive function over a typical short interval in terms of its long average. As part of this treatment, we use a variant of the Matom\"{a}ki-Radziwill theorem for divisor-bounded multiplicative functions recently proven by the author. We consider two sets of applications of these methods. Our first application shows that for an additive function $g: \mathbb{N} \rightarrow \mathbb{C}$ any non-trivial savings in the size of the average gap $|g(n)-g(n-1)|$ implies that $g$ must have a small first moment, i.e., the discrepancy of $g(n)$ from its mean is small on average. We also obtain a variant of such a result for the second moment of the gaps. This complements results of Elliott and of Hildebrand. As a second application, we make partial progress on an old question of Erd\H{o}s relating to characterizing $\log n$ as the only "almost everywhere" increasing additive function (up to constant factors). We show that if an additive function is almost everywhere non-decreasing then it is almost everywhere well-approximated by a constant times a logarithm. We also show that if $g$ is a completely additive function such that the density of the set of exceptions $\frac{1}{X}|\{n \leq X : g(n) < g(n-1)\}|$ decays like $O((\log X)^{-2-\epsilon})$ and such that $g$ is not extremely large too often on the primes (in a precise sense), then $g$ is identically equal to a constant times a logarithm.