Activity 1-15: Ergodic mathematics

What follows is an introduction to a newish branch of mathematics- ERGODIC mathematics. Task: how many irrational numbers do you know? Write down five. Now pick any number between 0 and 1, let’s call it t. Firstly -

Work out t + α, and then throw away everything except the decimal part (call this a 1 ). Now pick another number between 0 and 1, let’s call it α, with the condition that α is irrational.

Now add α again to this, throwing away everything except the decimal part once more (call this a 2.) Find a 1, a 2, a 3, a 4, a 5 and a 6. Can we say anything about the numbers a 1, a 2, a 3...? They will all be between 0 and 1: are some parts of (0, 1) more likely to be hit than others? It is a fundamental theorem of ergodic maths that all parts of (0, 1) will be hit equally often.

This can be explored using an Excel spreadsheet. Task: explore the Ergodic Spreadsheet.Spreadsheet

This fundamental ergodic theorem says: in the long run, the probability that this process ends up giving us a value in the interval (a, b), where 0 < a < b < 1, is b  a. Ergodic maths concerns itself with repeated processes. Here we had a map from one set to itself. This gave us the sequence a 1, a 2, a 3, a 4, a 5, a This sequence of points is called the orbit of a 1.

The nature of the orbit tells us a lot about α. If α is rational, the orbit is finite. If α is irrational, the orbit is infinite, and ‘equally spread’ on [0,1).

Another problem that classes as ergodic - we all know this sequence: 1, 2, 4, 8, 16, 32, 64, Consider this sequence, found by taking the starting digits from the above: 1, 2, 3, 8, 1, 3, 6, 1... Which digit occurs most often in this sequence?

 5  10 2  2 8 < (5 + 1)  Define {t} as ‘the fractional part of t’. Taking logs to base 10, log (5) + 2  8  log(2) < log (5 + 1) + 2.  log(5)  {8 log (2)} < log (5 + 1). Consider 500  512 < 600. Can we generalise this?

Now log(2) is irrational, and 0  log(k) < log (k + 1)  1. Consider k  10 j  2 n < (k + 1)  10 j. Taking logs to base 10, log (k) + j  n  log(2) < log (k + 1) + j.  log(k)  {n log (2)} < log (k + 1). So by our fundamental ergodic theorem, P(starting digit of 2 n = k) = log(k+1)  log (k) = log((k+1)/k).

So considering the values of k from {1, }, when is log(k + 1)/log(k) greatest? Clearly when k = 1. So 1 is the most common starting digit for powers of 2.

Here is another theorem that deserves the adjective ergodic: Task: find arithmetic progressions of length 3, 4, 5… by looking at the smallest prime numbers. Prime Arithmetic Progression Spreadsheet carom/carom-files/carom xlsm

Below we have the arithmetic sequences using the smallest possible numbers of the given length made up purely of prime numbers. What are the longest known arithmetic sequences made up of prime numbers?

On January 18, 2007, Jarosław Wróblewski found the first known case of 24 primes in arithmetic progression: 468,395,662,504, ,619  223,092,870  n, for n = 0 to 23. The constant here is the product of all the prime numbers up to 23.

On May 17, 2008, Wróblewski and Raanan Chermoni found the first known case of 25 primes: 6,171,054,912,832, ,384  223,092,870  n, for n = 0 to 24. On April 12, 2010, Benoãt Perichon (with software by Wróblewski and Geoff Reynolds) found the first known case of 26 primes: 43,142,746,595,714, ,681,770  223,092,870  n, for n = 0 to 25.

With thanks to: Manfred Einsiedler and Tom Ward, and their book, Introduction to Ergodic Theory. Wikipedia, for another excellent article. Graeme McRae, for his helpful site. Carom is written by Jonny Griffiths,