A Structure Theorem for Level Sets of Multiplicative Functions and Applications

Given a level set E of an arbitrary multiplicative function f, we establish, by building on the fundamental work of Frantzikinakis and Host [14, 15], a structure theorem that gives a decomposition of $1_{E}$ into an almost periodic and a pseudo-random part. Using this structure theorem together with the technique developed by the authors in [3], we obtain the following result pertaining to polynomial multiple recurrence.