**Fast track observations & questions after Liouville**

**Observation. In connection with the irrationality of (e.g.) **
** one comes to know that the important result that the equation **
** has an infinite number of solutions in (positive) integers **
**x**
**, **
**y**
**. One may think of that as starting from the meaning of **
**'s irrationality: there are no (positive) integers **
**x, y**
** such that **
**, and so the **
**next best**
** thing is to have **
** (or **
**) (An interested reader, not familiar with this, would benefit, I believe, from reading my web site notes on - what I call - **
**L- and R-approximations. **
**In general it's not just **
** that would be of interest, but **
**, and thus not just **
**, but **
**).**

** Every such **
**x**
**, **
**y**
** leads to a fantastic rational approximation **
** to **
**, as good an approximation, in fact, as there can possibly be. Slightly throwing away some of the quality of approximation, one has:**

** < **
** (infinitely often, in fact) (i)**

** **

** Now, inequality (i) happens to be (independently) gauranteed by the completely general:**

**Theorem (Dirichlet). Let **
** be any real irrational number (and it doesn't matter whether **
** is algebraic or transcendental), then there are an infinite number of rational numbers **
** (**
**p**
**, **
**q**
** integers, **
**q**
** > 0) such that**

** < **
** ... (ii)**

**Question. Are there any such **
**s for which a better approximation than (ii) could happen?**

**Answer. Yes, of course. It's easy. By simply varying the type of number encountered in the Liouville section, and forming a number **
**like**
** (e.g.) **
** then one obtains an **
** for which the inequality**

** < **
** ... (iii)**

**has an infinite number of solutions in rational numbers **
** (a careful reader will immediately spot that I haven't quite got the full validity of (iii), but **
**almost**
**, and**
** **
**it requires nothing more than than gorey extra detail to get the full validity...)**

**Another question. And how many such **
**s can one get?**

**Immediate answer, and observation. It's easy, and again - like I've already pointed out in the Liouville section - there are an uncountable numbers of such numbers: simply make up more **
**s like this **

**where the sequence {**
**} is chosen as in the Liouville section, and **
**etc**
**.**

** In fact, for any fixed **
** > 2, there are uncountably many real numbers **
** such that each of them has infinitely many rational approximations **
** satisfying**

** < **
** ... (**
**)**

**A Measure Theory motivated question. So, there are uncountably many **
**s satisfying the previous inequality, but how much space do they take up on the number line?**

**Immediate answer. Let **
** be the set of all real numbers for which inequality (**
**) has an infinite number of rational solutions, then [although **
** appears to be large, and certainly is from a cardinality point of view; in fact it has the same cardinality as the entire real line!] **
** has Lebesgue measuere zero [and so appears to be **
**quite small**
**].**

**Another question. Earlier it was observed that the equation **
** having an infinite number of solutions in (positive) integers **
**x**
**, **
**y **
**is a **
**natural outcome**
** of observing that **
** is irrational; what happens if one replaces (e.g.) the irrational number **
** with the irrational number **
**, and what then is the effect on the equation **
** (since there are no (positive) integers **
**x, y**
** such that **
**, and so the **
**next best**
** thing is to have **
**, i.e., **
**)**

**Every such **
**x**
**, **
**y **
**would create a rational number **
** **
**so close**
** to **
**as to **
**almost**
** be a solution of the inequality**

** < **
** ... (iii)**

**By Liouville's theorem all such **
**x**
**, **
**y**
** must satisfy the inequality**

** **
**>**
** **
** , for some constant **
**c**

**A big question then is: Does/doesn't the equation **
** (and others like it: **
**, general **
**non-cube**
** **
**d**
**, **
**, etc) have **
**any**
** solutions in integers **
**x**
**, **
**y**
**, and if so does it have infinitely many?**

**An answer. We are now getting into very, very deep water, and an answer will have to wait until we get to the Axel Thue section.**