A density on the natural numbers invariant with respect to the multiplication - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T09:37:45Z http://mathoverflow.net/feeds/question/59101 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/59101/a-density-on-the-natural-numbers-invariant-with-respect-to-the-multiplication A density on the natural numbers invariant with respect to the multiplication Valerio Capraro 2011-03-21T20:09:42Z 2011-03-22T23:08:20Z <p>The "classical Beurling density" of a subset of the natural numbers is $d(A)=lim_{n\rightarrow\infty}\frac{|A\cap[1,n]|}{n}$, when it exists. It defines a finitely additive probability measure on the natural numbers which is invariant with respect to the sum. Here is my question: does there exist a "nice formula" to describe a finitely additive probability measure on $\mathbb N$ which is invariant with respect to the multiplication? </p> <p>A couple of remarks: I don't know if it is trivial that such a measure exists, but anyway it follows from the application of a general result of Vern Paulsen (on arxiv "Syndetic sets and amenability"). Another problem would be that of finding the measure of particular sets. What about the measure of {$1!,2!,3!,4!...$}? Sets with measure differente from 0 and 1? For example the set of numbers whose first digit through 4 to 9 seems to have measure $=\log_{10}4$... any other?</p> <p>Thanks in advance, Valerio</p> http://mathoverflow.net/questions/59101/a-density-on-the-natural-numbers-invariant-with-respect-to-the-multiplication/59119#59119 Answer by Gerald Edgar for A density on the natural numbers invariant with respect to the multiplication Gerald Edgar 2011-03-21T23:50:05Z 2011-03-21T23:50:05Z <p>How about: $$\lim_{k\to\infty} \frac{|A \cap [1,k!]|}{k!}$$ when it exists?</p> http://mathoverflow.net/questions/59101/a-density-on-the-natural-numbers-invariant-with-respect-to-the-multiplication/59141#59141 Answer by Daniel Litt for A density on the natural numbers invariant with respect to the multiplication Daniel Litt 2011-03-22T05:29:45Z 2011-03-22T16:41:14Z <p>Here is an example. Let $d(A)$ be as you define it in the question, namely $$d(A)=\lim_{n\to \infty} \frac{|A\cap [1,n]|}{n}.$$ Let <code>$U_n=\{k\in\mathbb{N} \text{ such that } k \text{ is a mutiple of } n!\}$</code>. Then define $$\mu(A)=\lim_{n\to \infty} n!\cdot d(A\cap U_n).$$ Then I claim that $\mu$ is finitely additive and multiplicatively invariant. Finite additivity is obvious. For multiplicative invariance, note that for $s>k$, we have $(s+k)!\cdot d(kA\cap U_{s+k})=s!\cdot d(A\cap U_{s})$, unless I've screwed something up. </p> <p>EDIT: Note by the way that one can replace the limit in the definition with $d$ with the Cesaro mean, for example, giving a much broader class of sets with defined measure. For example, with this addition, the set of natural numbers with a fixed leading digit in a fixed prime base $p$ has density $1/(p-1)$. </p> http://mathoverflow.net/questions/59101/a-density-on-the-natural-numbers-invariant-with-respect-to-the-multiplication/59224#59224 Answer by Mark Schwarzmann for A density on the natural numbers invariant with respect to the multiplication Mark Schwarzmann 2011-03-22T19:30:34Z 2011-03-22T23:08:20Z <p>The natural thing to do here is to replace the intervals $[1,n]$ (and $n = |[1,n]|$) in the definition of $d(A)$ with a sequence $F_n$ of subsets of $\mathbb{N}$ which is multiplicatively asymptotically invariant (or, in other words, a Folner sequence for the semigroup $(\mathbb{N},\cdot)$). For an exploration of this idea, as well as applications, see for instance this article by Vitaly Bergelson:</p> <p><a href="http://www.math.osu.edu/~vitaly/vb_multiplic_23jul04.pdf" rel="nofollow">Multiplicatively large sets and ergodic Ramsey theory, Israel Journal of Mathematics 148 (2005), 23-40.</a></p> <p>EDIT: One particular example (mentioned in the article) is to take $F_n$ to be the set of all positive integers which can be written as a product of powers of the first $n$ primes, where the powers are allowed to be any non-negative integer which is less than or equal to $n$. The motivation for choosing $F_n$ this way is that just as $1$ generates the additive semigroup $(\mathbb{N},+)$, the primes generate the multiplicative semigroup. Think of balls in the corresponding Cayley graphs with radius getting larger and larger. The article contains many other examples of such $F_n$, and each of them gives a notion of "multiplicative density" by setting $d(A) = \lim_{n \to \infty} \frac{|A \cap F_n|}{|F_n|}$ (if the limit exists. Otherwise one usually considers the limsup and liminf).</p>