Finding an Irrational not in the Union.

[Bacle]

Hi, All:

This is an old problem I never solved, and I recently saw somewhere else:

We are given an enumeration {q_1,q_2,..,q_n,...} of the rationals in the real line.

We construct the union : S:=\/ (q_i+[e/2^(i+1)] , q_i-[e/2^(i+1)] ) for i=1,2..,n,..

i.e., we want the total measure of S to be a real eps>0; a fixed real number.

Question: given a choice of enumeration, we want to find an irrational that is _not_

in S. I think the best idea is using Roth-Siegel approximation and somehow

using an algebraic number that cannot be approximated too well by rationals with

fixed denominator, but I can't think of a full solution. Any Ideas?

number that cannot

 

[jvicens]

be approximated too well by rationals with fixed denominator.

Hello,

Thank you for bringing up this interesting problem. First, let me clarify the notation used in the forum post. The symbol "\/" represents the union operation, so S is the union of all the intervals (q_i+[e/2^(i+1)], q_i-[e/2^(i+1)]) for i=1,2,...,n,... The goal is to find an irrational number that is not contained in this union, regardless of the choice of enumeration.

One approach to finding such an irrational number is to use the concept of a "well-approximable" number. A real number x is said to be well-approximable if there exist infinitely many rational numbers p/q such that |x-p/q| < 1/q^2. In other words, x can be approximated very well by rational numbers with increasingly large denominators.

Now, if we can find an irrational number x that is not well-approximable, then it will not be contained in the union S. This is because for any given rational number p/q, the difference |x-p/q| cannot be smaller than 1/q^2. Therefore, x will always be outside the intervals (p/q+[e/2^(q+1)], p/q-[e/2^(q+1)]) for any choice of enumeration.

One way to construct such an irrational number is to use the continued fraction expansion. If we have a real number x with a continued fraction expansion [a_0;a_1,a_2,...], where a_i are the digits in the expansion, then we can define a sequence of rational approximations p_n/q_n as follows:

p_0/q_0 = a_0
p_1/q_1 = a_0 + 1/a_1
p_2/q_2 = a_0 + 1/(a_1 + 1/a_2)
and so on.

It can be shown that for any irrational number x, the sequence p_n/q_n will always converge to x. However, if x is not well-approximable, then this convergence will be very slow, meaning that for any given rational number p/q, the difference |x-p/q| will not decrease quickly enough to be smaller than 1/q^2.

Therefore, to find an irrational number that is not well-