A dynamical point of view on the set of B-free integers

By el Houcein el Abdalaoui, Mariusz Lemanczyk, Thierry de la Rue. 2013

70 citations recorded by the source corpus.

We extend the study of the square-free flow, recently introduced by Sarnak, to the more general context of B-free integers, that is to say integers with no factor in a given family B of pairwise relatively prime integers, the sum of whose reciprocals is finite. Relying on dynamical arguments, we prove in particular that the distribution of patterns in the characteristic function of the B-free integers follows a shift-invariant probability measure, and gives rise to a measurable dynamical system isomorphic to a specific minimal rotation on a compact group. As a by-product, we get the abundance of twin B-free integers. Moreover, we show that the distribution of patterns in small intervals also conforms to the same measure. When elements of B are squares, we introduce a generalization of the Mobius function, and discuss a conjecture of Chowla in this broader context.

Subfields: Artificial Intelligence, Mathematical Physics, Theoretical Computer Science

View Paper: https://www.semanticscholar.org/paper/ac00978153e47c60ab95fbc88599a51c0a31dbfb

How researchers see this paper

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In Artificial Intelligence

1. J. Kułaga-Przymus
2. Mariusz Lemanczyk

In Mathematical Physics

1. J. Kułaga-Przymus
2. Mariusz Lemanczyk
3. el Houcein el Abdalaoui
4. Thierry de la Rue
5. Terence Tao

In Theoretical Computer Science

1. J. Kułaga-Przymus
2. Mariusz Lemanczyk

Citations in the corpus

• The Chowla and the Sarnak conjectures from ergodic theory point of view (2014, methodology): “5 Möbius function and its generalizations The following generalization of the Möbius function μ : N∗ → {−1, 0, 1} defined by (1) has been introduced in [2].”