Additive functions in short intervals, gaps and a conjecture of Erdős

Abstract

With the aim of treating the local behaviour of additive functions, we develop analogues of the Matomäki–Radziwiłł theorem that allow us to approximate the average of a general additive function over a typical short interval in terms of a corresponding long average. As part of this treatment, we use a variant of the Matomäki–Radziwiłł theorem for divisor-bounded multiplicative functions recently proven in Mangerel (Divisor-bounded multiplicative functions in short intervals. arXiv: 2108.11401). We consider two sets of applications of these methods. Our first application shows that for an additive function {varvec{g:}} mathbb {N} rightarrow mathbb {C} any non-trivial savings in the size of the average gap |{varvec{g}}{} {textbf {(}}{varvec{n}}{} {textbf {)}}-{varvec{g}}{} {textbf {(}}{varvec{n}}-{textbf {1}}{} {textbf {)}} | implies that {varvec{g}} must have a small first centred moment i.e. the discrepancy of {varvec{g}}{} {textbf {(}}{varvec{n}}{} {textbf {)}} 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 Erdős relating to characterizing constant multiples of {{textbf {log}}} ,{varvec{n}} as the only almost everywhere increasing additive functions. 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 the set {{varvec{n}} in mathbb {N} : {varvec{g}}{} {textbf {(}}{varvec{n}}{} {textbf {)}} < {varvec{g}}{} {textbf {(}}{varvec{n}}-{textbf {1}}{} {textbf {)}}} is sufficiently sparse, and if {varvec{g}} is not extremely large too often on the primes (in a precise sense), then {varvec{g}} is identically equal to a constant times a logarithm.