2015/09/26 by Aurelia Bartnicka, Stanisław Kasjan, Bartnicka, Aurelia +5 · 2 citations
Mathematics · Physics and Astronomy · #Analytic Number Theory Research #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #math.DS
paper · pdf · doi:10.48550/arxiv.1509.08010
79 pages
arxiv created 2015/09/26 · openalex publication_date 2015/09/26 · arxiv updated 2015/09/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
Let B⊂ ℕ and let η∈ \0,1\^ℤ be the characteristic function of the set FB:=ℤ∖\bigcupbbℤ of B-free numbers. Consider (S,Xη), where Xη is the closure of the orbit of η under the left shift S. When B=\p2 : p∈ P\, (S,Xη) was studied by Sarnak. This case + some generalizations, including the case (*) of B infinite, coprime with ∑b1/b<∞, were discussed by several authors. For general B, contrary to (*), we may have Xη\subsetneq XB:=\x∈ \0,1\^ℤ : |supp x\bmod b|≤ b-1 ∀b\. Also, Xη may not be hereditary (heredity means that if x∈ X and y≤ x coordinatewise then y∈ X). We show that η is quasi-generic for a natural measure νη. We solve the problem of proximality by showing first that Xη has a unique minimal (Toeplitz) subsystem. Moreover B-free system is proximal iff B contains an infinite coprime set. B is taut when δ(FB)<δ(F_B∖ \b\ ) for each b. We give a characterization of taut B in terms of the support of νη. Moreover, for any B there exists a taut B' with νη=νη'. For taut sets B,B', we have B=B' iff XB=XB'. For each B there is a taut B' with Xη'⊂ Xη and all invariant measures for (S,Xη) live on Xη'. (S,Xη) is shown to be intrinsically ergodic for all B. We give a description of all invariant measures for (S,Xη). The topological entropies of (S,Xη) and (S,XB) are both equal to d(FB). We show that for a subclass of taut B-free systems proximality is the same as heredity. Finally, we give applications in number theory on gaps between consecutive B-free numbers. We apply our results to the set of abundant numbers.