Special cases and equivalent forms of Katznelson's problem on recurrence

DOI10.1007/S00605-022-01770-6arXiv2108.02190MaRDI QIDQ6374518

Publication date: 4 August 2021

Abstract: We make three observations regarding a question popularized by Katznelson: is every subset of

m a t h b b Z

which is a set of Bohr recurrence is also a set of topological recurrence? (i) If

G

is a countable abelian group and

E s u b s e t G

is an

I_{0}

set, then every subset of

E - E

which is a set of Bohr recurrence is also a set of topological recurrence. In particular every subset of

2^{n} - 2^{m} : n, m i n m a t h b b N

which is a set of Bohr recurrence is a set of topological recurrence. (ii) Let

m a t h b b Z^{o m e g a}

be the direct sum of countably many copies of

m a t h b b Z

with standard basis

E

. If every subset of

(E - E) - (E - E)

which is a set of Bohr recurrence is also a set of topological recurrence, then every subset of every countable abelian group which is a set of Bohr recurrence is also a set of topological recurrence. (iii) Fix a prime

p

and let

m a t h b b F_{p}^{o} m e g a

be the direct sum of countably many copies of

m a t h b b Z / p m a t h b b Z

with basis

(m a t h b f e_{i})_{i i n m a t h b b N}

. If for every

p

-uniform hypergraph with vertex set

m a t h b b N

and edge set

m a t h c a l F

having infinite chromatic number, the Cayley graph on

m a t h b b F_{p}^{o} m e g a

determined by

s u m_{i i n F} m a t h b f e_{i} : F i n m a t h c a l F

has infinite chromatic number, then every subset of

m a t h b b F_{p}^{o} m e g a

which is a set of Bohr recurrence is a set of topological recurrence.

Mathematics Subject Classification ID

Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.) (43A46) Notions of recurrence and recurrent behavior in topological dynamical systems (37B20)

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