Florian Karl Richter

A Structure Theorem for Level Sets of Multiplicative Functions and Applications

Given a level set

Rationally almost periodic sequences, polynomial multiple recurrence and symbolic dynamics

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On the Density of Coprime Tuples of the Form ( n , ⌊ f 1 ( n )⌋, …, ⌊ f k ( n )⌋), Where f 1 , …, f k Are Functions from a Hardy Field

of an arbitrary multiplicative function

Large subsets of discrete hypersurfaces in Z d contain arbitrarily many collinear points

f

On the density of coprime tuples of the form

, we establish, by building on the fundamental work of Frantzikinakis and Host [13,14], a structure theorem which gives a decomposition of

(n,\lfloor f_1(n)\rfloor,\ldots,\lfloor f_k(n)\rfloor)

\mathbb{1}_E

, where

into an almost periodic and a pseudo-random parts. Using this structure theorem together with the technique developed by the authors in [3], we obtain the following result pertaining to polynomial multiple recurrence. Let

f_1,\ldots,f_k

E=\{n_1<n_2<\ldots\}

are functions from a Hardy field

be a level set of an arbitrary multiplicative function with positive density. Then the following are equivalent: -

Multiplicative combinatorial properties of return time sets in minimal dynamical systems.

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