A Structure Theorem for Level Sets of Multiplicative Functions and Applications

DOI10.1093/IMRN/RNY040zbMATH Open1447.37009arXiv1708.02613OpenAlexW2743434280WikidataQ131317257 ScholiaQ131317257MaRDI QIDQ4960124

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Publication date: 8 April 2020

Published in: (Search for Journal in Brave)

Abstract: Given a level set

E

of an arbitrary multiplicative function

f

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

m a t h b b 1_{E}

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

E = n_{1} < n_{2} < l d o t s

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

E

is divisible, i.e. the upper density of the set

E c a p u m a t h b b N

is positive for all

u i n m a t h b b N

; -

E

is an averaging set of polynomial multiple recurrence, i.e. for all measure preserving systems

(X, m a t h c a l B, m u, T)

, all

A i n m a t h c a l B

with

m u (A) > 0

, all

e l l g e q 1

and all polynomials

p_{i} i n m a t h b b Z [x]

,

i = 1, l d o t s, e l l

, with

p_{i} (0) = 0

we have

lim_{N oinfty}frac{1}{N}sum_{j=1}^N mu�ig(Acap T^{-p_1(n_j)}Acapldotscap T^{-p_ell(n_j)}A�ig)>0.

We also show that if a level set

E

of a multiplicative function has positive upper density, then any self-shift

E - r

,

r i n E

, is a set of averaging polynomial multiple recurrence. This in turn leads to the following refinement of the polynomial Szemer'edi theorem (cf. [4]). Let

E

be a level set of an arbitrary multiplicative function, suppose

E

has positive upper density and let

r i n E

. Then for any set

D s u b s e t m a t h b b N

with positive upper density and any polynomials

p_{i} i n m a t h b b Q [t]

,

i = 1, l d o t s, e l l

, which satisfy

p_{i} (m a t h b b Z) s u b s e t m a t h b b Z

and

p_{i} (0) = 0

for all

i i n 1, l d o t s, e l l

, there exists

such that the set

left{,nin E-r:overline{d}Big(Dcap (D-p_1(n))cap ldotscap(D-p_ell(n)) Big)>�eta , ight}

has positive lower density.

Full work available at URL: https://arxiv.org/abs/1708.02613

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