Inverse problem for upper asymptotic density

Jin, Renling

doi:10.1090/S0002-9947-02-03122-7

Inverse problem for upper asymptotic density
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Trans. Amer. Math. Soc. 355 (2003), 57-78 Request permission

Abstract:

For a set $A$ of natural numbers, the structural properties are described when the upper asymptotic density of $2A+\{0,1\}$ achieves the infimum of the upper asymptotic densities of all sets of the form $2B+\{0,1\}$, where the upper asymptotic density of $B$ is greater than or equal to the upper asymptotic density of $A$. As a corollary, we prove that if the upper asymptotic density of $A$ is less than $1$ and the upper asymptotic density of $2A+\{0,1\}$ achieves the infimum, then the lower asymptotic density of $A$ must be $0$.

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