Publication

On multiplicative recurrence along linear patterns

2025

Abstract

Donoso, Le, Moreira, and Sun (J. Anal. Math. 149 (2023), 719–761) study sets of recurrence for actions of the multiplicative semigroup and provide some sufficient conditions for sets of the form to be sets of recurrence for such actions. A necessary condition for to be a set of multiplicative recurrence is that for every completely multiplicative function taking values on the unit circle, we have that . In this article, we fully characterize the integer quadruples which satisfy the latter property. Our result generalizes a result of Klurman and Mangerel (Math. Ann. 372 (2018), 651–697) concerning the pair , as well as some results from (Donoso, Le, Moreira, and Sun, J. Anal. Math. 149 (2023), 719–761). In addition, we prove that, under the same conditions on , the set is a set of recurrence for finitely generated actions of .

Official source

https://infoscience.epfl.ch/entities/publication/b44dd0b8-b17c-4f1f-9683-19e83fff865e

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2025

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/b44dd0b8-b17c-4f1f-9683-19e83fff865e

About this result

Related concepts (32)

Semigroup

In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily the elementary arithmetic multiplication): x·y, or simply xy, denotes the result of applying the semigroup operation to the ordered pair (x, y). Associativity is formally expressed as that (x·y)·z = x·(y·z) for all x, y and z in the semigroup.

Transformation semigroup

In algebra, a transformation semigroup (or composition semigroup) is a collection of transformations (functions from a set to itself) that is closed under function composition. If it includes the identity function, it is a monoid, called a transformation (or composition) monoid. This is the semigroup analogue of a permutation group. A transformation semigroup of a set has a tautological semigroup action on that set. Such actions are characterized by being faithful, i.e., if two elements of the semigroup have the same action, then they are equal.

Finite group

In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. Important examples of finite groups include cyclic groups and permutation groups. The study of finite groups has been an integral part of group theory since it arose in the 19th century.