Stowaway.html

The Stowaways

There is a familiar yet amazing proof that the measure of the rationals is zero. One enumerates the rationals r₀, r₁, r₂, ... and puts an open interval of length 2⁻ⁿ about r_n. The union, u₀, of these intervals is an open set whose measure is no more than 2. One then proceeds to cut each interval in half, calling their union u₁; the measure of u₁ is less than 1. Continuing this to produce u₂, u₃, ... one sees that the measure is less than any positive number and thus is zero.

One might guess that the limit of these ever smaller open sets would be the rationals alone, but there are stowaway irrationals included in the limit; where they hide depends on the enumeration. Different enumerations will result in different stowaways. Here is a construction of c such stowaways within any interval.

First I will describe a general construction to produce c reals. It can also be used to build the Cantor set. Consider a sequence of closed intervals i₁, i₂, i₃ ... such that intervals i_2j and i_2j+1 are disjoint sub intervals of i_j. Assume that the length of these intervals converges to zero. One can choose a sub sequence of intervals starting from i₁ and at each step taking either i_2j or i_2j+1 as the interval following i_j. This sub sequence will form a convergent Cauchy sequence of nested intervals that must converge to some real. There are c such sub sequences, each determined by a sequence of choices. Each sub sequence leads to a different real.

The cantor set can be produced by letting i₁ be [0, 1] and for each j letting i_2j be the left most third of i_j and i_2j+1 be the right most third.

Now we adapt this procedure to find the stowaways. That they are not all rational follows from the fact that there are too few rationals.

At stage k of halving the intervals, we refer to the interval about r_n whose length is 2^−(n+k) as i(n, k).

Consider the following recursive routine that takes an open interval I and a stage index k:

choose a rational r_n in I.
Choose a closed sub-interval j of (i(n, k) ∩ I)
Pick distinct rationals r_p and r_q from j.
Perform this routine for i(p, k+1) ∩ j and i(q, k+1) ∩ j for stage index k+1.

The resulting closed intervals form c Cauchy interval sequences which each converge to a distinct real. Each of these limit reals is a stowaway.

The successive closed intervals of each Cauchy sequence are taken from the successive iterations of the routine. At each iteration one chooses between two sub intervals. There are c such possible choice sequences.

At each stage k there is a union u_k of intervals which covers the rationals. At each stage the complement C_k of u_k is nowhere dense since u_k covers the rationals. The complement of u_k is nowhere dense. The union of these complements is thus a denumerable union of nowhere dense sets. Such a set is called meagre.