7
$\begingroup$

Strauch & Tóth [1] Georges Grekos [3][4] showed that for any choice of upper and lower density, there is some subset of $\mathbb{N}$ with the chosen densities, provided the lower is no more than the upper and both are in [0, 1].

Mišík [2] extended this to show that for any choice of upper density, lower density, upper logarithmic density, and lower logarithmic density there is a subset of $\mathbb{N}$ with the chosen densities, provided they follow the necessary inequalities $0\leqslant\underline d\leqslant\underline\delta\leqslant\overline\delta\leqslant\overline d\leqslant 1.$

Is there a similar result with the uniform densities? $$ \underline{u}(A)=\lim_s\frac1s\liminf_n\sum_\stackrel{a\in A}{\ n < a\leqslant n+s}1 $$ $$ \overline{u}(A)=\lim_s\frac1s\limsup_n\sum_\stackrel{a\in A}{\ n < a\leqslant n+s}1 $$

An ideal result would combine all three density types with the inequality $$ 0\leqslant\underline{u}\leqslant\underline{d}\leqslant\underline{\delta}\leqslant\overline{\delta}\leqslant\overline{d}\leqslant\overline{u}\leqslant1 $$ but I'm looking for any published results on the topic.

[1] Strauch and Tóth, Asymptotic density of $A\subset N$ and density of the ratio set $R(A)$. Acta Arith. 87 (1998), pp. 67-78. eudml

[2] Ladislav Mišík, Sets of positive integers with prescribed values of densities, Mathematica Slovaca 52:3 (2002), pp. 289-296.dml.cz

[3] Georges Grekos, Sur la répartition des densités des sous-suites d'une suite donnée, PhD thesis, Université Pierre et Marie Curie (1976).

[4] Georges Grekos, Répartition des densites des sous-suites d'une suite d'entiers, J. Number Theory 10:2 (1978), pp. 177-191 (in French). doi:10.1016/0022-314X(78)90034-3

$\endgroup$
  • 1
    $\begingroup$ I have no idea about the literature, but this shouldn't be too hard to construct by hand. Certainly it's easy to up $\bar u$ and lower $\underline u$ without affecting any of the others: very occasionally (e.g. introduce a segment of length $n$ at location $e^{e^n}$) introduce lengthening segments with very uniform density $\bar u$ and $\underline u$ (e.g. obtained from an irrational rotation). This would mean you're done provided you start off with a sequence where $\underline u=\underline d$ and $\bar d=\bar u$. Thus provided Mi\v sik satisfies this, there's nothing more to check. $\endgroup$ – Anthony Quas Jul 25 '12 at 20:04
  • 2
    $\begingroup$ @Charles: AFAIK, the result that you credit to Strauch & Toth was 1st proved by Georges Grekos in his thesis (Paris 6, 22 June 1976), and later published in: G. Grekos, Répartition Des Densites Des Sous-Suites D'Une Suite D'Entiers, J. Number Theory 10 (1978), No. 2, 177-191 (in French). Strauch & Tóth's paper is definitely focused on a different problem (what's the minimal lower asymptotic density of a set $X\subseteq\bf N^+$ for which the ratio set of $X$, viz. $\{x/y: x,y\in X\}$, is dense in $[0,\infty[$?), and recovers Grekos' result as a corollary. $\endgroup$ – Salvo Tringali May 20 '15 at 18:51
  • 3
    $\begingroup$ @KevinO'Bryant: I join Charles in his request. What are you alluding to? I don't claim to have read Mišík's paper carefully, but as far as I can say, there is no problem in there (incidentally, the paper is cited, e.g., in: F. Luca, C. Pomerance, and Š. Porubský, Sets with prescribed arithmetic densities, Unif. Distr. Theory 3 (2008), No. 2, 67-80, where no reference is made to any gap or other issue in Mišík's work). $\endgroup$ – Salvo Tringali May 20 '15 at 19:01
  • 1
    $\begingroup$ @KevinO'Bryant: Mišík's paper has two main results, Th. 1 (p. 291) & Th. 2 (p. 293): Th. 1 is about the independence of the lower and upper $f$-densities, ${\underline{d}}_f$ and ${\overline{d}}^f$ in Mišík's notation, associated with a function $f:{\bf N}^+\to{\bf R}^+$ s.t. $\sum_{n\ge 1}f(n)=\infty$ and $f(n)/\sum_{i=1}^nf(i)\to 0$ as $n\to\infty$, while Th. 2 is about the independence of the upper and lower asymptotic and logarithmic densities. In fact, the proof of Th. 2 is just 2+1/2 pages, since we don't really need Lemma 1 here (even if this is not mentioned in the paper...). [tbc] $\endgroup$ – Salvo Tringali May 24 '15 at 5:03
  • 1
    $\begingroup$ However, this doesn't mean much to me, when compared with the length of the proof of the main theorem (viz., Th. 3 on p. 77) in Luca & Porubský's 2005 paper, since the latter is, to my eyes, much stronger (and more constructive) than Mišík's Th. 2. Again, I've not read Mišík's paper carefully (I should, but had no time). Yet, I mentioned your comment to other guys who did it (I believe!), and they replied that they aren't aware of any issue with Mišík's paper. So would you mind to be more specific on what you think is missing in Mišík's proof? I'm very interested, and other peps may be too. $\endgroup$ – Salvo Tringali May 24 '15 at 5:26
5
$\begingroup$

OK. I think you can do this pretty easily by hand. First you need a way to generate sequences with very uniform density. The Sturmian sequences are perfect for this. A Sturmian sequence with parameter $\alpha$ has the property that sub-blocks of length $N$ have density converging to $\alpha$ uniformly in $N$.

Now: start by interspersing a Sturmian with parameter $\underline\delta$ with one of parameter $\bar\delta$. How to do the interspersing? Have one from $2^{n!}$ to $2^{(n+1)!}$. Then switch to the other between $2^{(n+1)!}$ and $2^{(n+2)!}$ etc. This switching is slow enough to guarantee that the sequence that you obtain has the prescribed $\underline\delta$ and $\bar\delta$. These are also the upper and lower densities for the time being.

Next we'll modify the sequence to obtain densities $\underline d$ and $\bar d$. Alternately splice in segments of Sturmian parameter $\underline d$ and $\bar d$ between $2^{4^i}$ and $2^{4^i+2^i}$. This won't affect the upper and lower logarithmic densities (because $2^i/4^i\to 0$).

But looking at these segments, you obtain a sequence with upper and lower densities $\underline d$ and $\bar d$ (the upper and lower uniform densities are also $\underline d$ and $\bar d$).

Finally, we'll perturb things as in my comment to get the uniform densities we want. For the segments between $2^n$ and $2^n+n$, insert alternately segments of the Sturmian sequences with densities $\underline u$ and $\bar u$. These segments are so sparse, they will have no effect on the upper and lower densities or the logarithmic densities. They are enough to guarantee that you get the uniform densities you want.

And Robert's your mother's brother.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.