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Notes on functions preserving density. (English) Zbl 1271.11010

For a set \(A,\) subset of the set \(\mathbb N\) of positive integers, one defines its upper (resp. lower) asymptotic density \({\overline d}(A)\) (resp. \({\underline d}(A)\)) as the upper (resp. lower) limit, as \(n\) tends to infinity, of \(n^{-1} |A\cap[1,n]|.\) If \({\overline d}(A)={\underline d}(A),\) this common value is called asymptotic density of \(A\) and is denoted by \(d(A).\)
Let \(\delta\) be a nonnegative real number. Denote by \({\mathcal D}_\delta\) the family of all sets \(A\) of positive integers such that \[ {\overline d}(A)-{\underline d}(A)\leq\delta. \] The authors prove the following theorems.
Theorem 1. Let \(h:{\mathbb N}\rightarrow{\mathbb N}\) be a function (not necessarily an one-to-one function) such that if \(A\in{\mathcal D}_0,\) then \(h(A)\in{\mathcal D}_0.\) Let \(\lambda= d(h({\mathbb N})).\) Then \(d(h(A))=\lambda d(A)\) for all \(A\in{\mathcal D}_0.\)
Theorem 2. Let \( \delta>0.\) Let \(f:{\mathbb N}\rightarrow{\mathbb N}\) be an one-to-one function such that if \(A\in{\mathcal D}_0\) then \(f(A)\in{\mathcal D}_\delta.\) Consider two sets \(A,B\) in \({\mathcal D}_0\) such that \(d(A)=d(B).\) Then \({\overline d}(B)-{\underline d}(A)\leq\delta.\)
From the above theorems, the authors deduce as a corollary a result proved in [M. B. Nathanson and R. Parikh, J. Number Theory 124, No. 1, 151–158 (2007; Zbl 1128.11006)] on functions preserving asymptotic density.

MSC:

11B05 Density, gaps, topology

Citations:

Zbl 1128.11006
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