This is old news by now but re that question about density one half, (A) Dan's proof is fine but a more general statement has a proof which is almost two-liner and doesn't need anything as deep as Lebesgue density. A couple of people C. Gardner and Y. Peres pointed out the following: _____________________________________________________________________ I write A(int)B for intersection, and m(S) for outer measure of S. THEORM.There is no set S (measurable or not) in [0,1] with the property that m(S(int)[a,b]) = (b-a)/2 for all intervals [a,b] in[0,1]. FACT; Every open set in R is a countable union of DISJOINT open intervals. Proof of Theorem. Assume S exist. Then m(S)=1/2, so there is an open set O containing S with 1/2<m(O) < 2/3. Using FACT, O is union of disjoint intervals J (where J will stand for the interval and also its length), but by assumption m(S(int)J)= J/2 so by subadditivity of outer measure m(S)<Summation(J/2) < 1/3<1/2, contradiction. At 09:42 PM 4/18/2005, you wrote:

1. A point of density of a set is a point such that for small enough neighborhoods, the density of the set in that neighborhood is arbitrarily close to 1. I.e. every neighborhood smaller than delta has 99.99% of points in the set, etc. The Lebesgue density theorem asserts that almost every element of any measurable set is a point of density. I.e. for any measurable subset of R (or any other space) almost every point has neighborhoods with density approaching 1 or density approaching 0. This is a fundamental principle of measure theory, I think it is (or at least it should be) in the basic texts. It really comes from the way measure is defined.

2. You can take an irrational rotation of the circle R/Z like x -> x + sqrt(2). The orbits of this rotation are all isomorphic to Z --- i.e. points never return to themselves. Pick a point from each orbit, using the axiom of choice. Partition the orbit into the even images and the odd images of this point. The union of all odd points in all orbits is "half" the circle in the sense that its congruent by a translation to its complement. In fact you can take odd powers of the rotation arbitrarily close to the identity, so you can make the set match its complement by an arbitrarily small rotation. This example is covered by an example in R --- you could think of this as taking any action of Z^2 (or Z^k would also work) by translations on R with dense orbits, then partitioning the orbits into cosets of a subgroup of index 2 acting on a selected point.

It's known that there's no definition or rule or humanly comprehensible procedure to actually make such a selection. Bill On Apr 18, 2005, at 9:24 PM, Michael Kleber wrote:

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Dan Asimov wrote:

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<< Can someone give me an example of a set of density 1/2 on every interval of the real numbers?

If such a set must be measurable, then no such set exists.

On the other hand, there do exist partitions of the reals into two dense homogeneous subsets that are related by a translation.

Probably I should know why both of these are true, but right now it seems that I don't. Can you explain?

--Michael Kleber

ps: "all reals whose last digit is even" :-?

-- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.

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