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

TL;DR

This paper develops short-interval analogues of the Matomäki-Radziwill theorem for additive functions, providing new insights into their local behavior, gap structure, and a partial resolution of Erdős's conjecture on almost everywhere increasing additive functions.

Contribution

It introduces short-interval approximations for additive functions, linking local averages to global behavior, and advances understanding of their gap distributions and monotonicity properties.

Findings

Small average gaps imply small first moments of additive functions.

Almost everywhere non-decreasing additive functions are approximated by logarithms.

Under certain conditions, additive functions are exactly proportional to logarithms.

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.