Timeline for Density of a set of natural numbers whose differences are not bounded.
Current License: CC BY-SA 3.0
11 events
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Jul 25, 2012 at 16:56 | comment | added | Charles | @Andreas: Densities are 11B05 in the 2000 MSC, so the AMS seems to think it's number theory. | |
May 28, 2011 at 11:55 | vote | accept | Valerio Capraro | ||
May 28, 2011 at 0:23 | comment | added | David Hansen | An explicit negative example: the positive integers, with all the integers in intervals of the form $[n^2,n^2+n]$ deleted. This has density one. | |
May 27, 2011 at 16:14 | answer | added | Roland Bacher | timeline score: 11 | |
May 27, 2011 at 15:15 | answer | added | Martin Sleziak | timeline score: 12 | |
May 27, 2011 at 14:28 | comment | added | Martin Sleziak | As an addendum to my previous comment: $\overline u(A)=0$ is equivalent to the condition that all shift-invariant means have the value zero on $A$. (The upper Banach density $\overline u(A)$ is the supremum/maximum of the values of shift-invariant means $\mu(A)$.) | |
May 27, 2011 at 14:22 | comment | added | Martin Sleziak | I believe that the condition that a set has arbitrary large gaps is equivalent to $\underline u(A)=0$, where $\underline u(A)$ stands for lower uniform density (a.k.a. lower Banach density). Relation to asymptotic density: $\underline u(A)\le \underline d(A) \le \overline d(A) \le \overline u(A)$. | |
May 27, 2011 at 14:10 | comment | added | Gjergji Zaimi | Possible counterexample: Choose a sparse enough set of primes with sum of reciprocals $<1$ and then consider the integers which are not divisible by any of these primes. | |
May 27, 2011 at 14:10 | comment | added | Andreas Blass |
A minor modification of Francois Brunault's solution gives you a partition of $\mathbb N$ into two pieces both of which have unbounded $d_n$ 's; where one piece has a big gap to make its $d_n$ big, the other has a run of consecutive 1's. That prevents any non-trivial mean (invariant or not) from being 0 on all such sets. (I suspect that number theorists will object to the "number theory" tag.)
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May 27, 2011 at 13:59 | comment | added | François Brunault | Regarding your first question, I think it's false : take an infinite sequence of consecutive integers, with only very rare jumps. | |
May 27, 2011 at 13:50 | history | asked | Valerio Capraro | CC BY-SA 3.0 |