Fall 2002: Diophantine Approximations

The aim of the theoretical side of this course is to familiarize the students with the foundations of Diophantine analysis. The starting point of the course is the classical theory of continued fractions. We will work on the theory of continued fractions until we have enough motivation to move on to the more delicate topic of "approximations by rational numbers;" we will affectionately refer to this topic as Diophantine Approximations. It is an easy exercise that a rational number cannot be approximated by other rationals to degree more than one. These notions will be made precise by the middle of the semester; to follow this account, all you need to keep in mind is that a number that can be approximated to degree 5 is better approximated than a number that can be approximated to degree 3, 7 is better than 2, etc. Also one can see -and this is a non-trivial fact- that any irrational number is approximateable by rational numbers to degree at least two. Then there is the classical theorem of Liouville that states that an algebraic number of degree n cannot be approximated to degree more than n. Well, this rocks! First of all it gives us an easy way to create transcendental numbers (how? you'll find out later), and second, it creates lots of good questions! For example, one question is that suppose an algebraic number A of degree n is given. Then what's the best degree to which we can approximate A by rational numbers? Remember, Liouville's theorem says this degree is not larger than n, but is that the best bound? The answer is that the best degree is indeed 2! (The ! ending the sentence is both an exclamation mark and a factorial sign, after all 2!=2, isn't it?) This is Roth's theorem. Don't try proving this at home, it's somewhat dangerous. After all somebody won a Fields medal for it... This statement we will prove, and more stuff if we have time.  

On the experimental side, students will numerically investigate interesting open problems in these fields. For an example of a past project which can easily be continued, see Robert Lipshitz's report.