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Dec 14, 2019 at 19:33 comment added combimateur Yes, I thought about the density.
Dec 14, 2019 at 17:10 comment added Anthony Quas For the natural numbers starting with 7, $d^*(A)=1$. This is upper Banach density, the density in long sub-intervals; not the density in initial sub-intervals.
Dec 14, 2019 at 10:48 comment added combimateur Great. Many Thanks. In fact this is not true that $d^*(A+\{1,\cdots, L\})\rightarrow 1$ when $L$ goes to infinity for $d^*(A)>0$ (take the natural numbers starting with a $7$...) so that I first believed my property was false.
Dec 14, 2019 at 6:34 history edited Anthony Quas CC BY-SA 4.0
edited body; added 6 characters in body
Dec 14, 2019 at 5:25 history answered Anthony Quas CC BY-SA 4.0