As a sophomore math student at ETH, I got a copy of Chinchin's "Drei
Perlen der Zahlentheorie":
The van der Waerden Theorem is:
If the natural numbers is partitioned
into finitely many sets, one of them contains an arbitrary long
arithmetic sequence.
For example, if we look at the set A of integers for which the n'th digit of π is
even and the set B of integers for which the n'th digit of π is odd, then
one of the two sets contains an arbitrary long arithmetic sequence. (The proof
does not tell which one!). A natural bet would be that both do but we do not know!
Lets look at this example with the computer
f[x_]:=If[EvenQ[x],1,0]; M=100;
P=IntegerDigits[Round[10^M*Pi]];
A=Flatten[Position[Map[f,P],1]]
B=Flatten[Position[Map[f,P],0]]
The set of n where the digit is even is
A={3, 7, 8, 12, 17, 19, 20, 21, 22, 23, 24, 27, 29, 33, 34, 35, 36,
37, 42, 51, 53, 54, 55, 58, 60, 61, 64, 66, 68, 70, 71, 72, 73, 74, 75,
76, 77, 78, 79, 82, 83, 84, 85, 86, 88, 89, 90, 93, 94, 98, 99, 100, 101}
The set where it is odd is
B={1, 2, 4, 5, 6, 9, 10, 11, 13, 14, 15, 16, 18, 25, 26, 28, 30, 31,
32, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 52, 56, 57, 59, 62,
63, 65, 67, 69, 80, 81, 87, 91, 92, 95, 96, 97}
I had been very impressed by the proof of Chinchine and later even more
mesmerized by the proof of Furstenberg (whose book was one of my bibles when
getting into dynamical systems theory). I mentioned the theorem in on page 376
of my thesis [PDF].
A Zd dynamical system is defined by d commuting homeomorphisms
T1,...,Td. A point x is called multiple recurrent if for
every open neighborhood U of x, there exists n such that T1n(x),
...,Tdn(x) are all in U. Here is the Fuerstenberg recurrent
theorem:
Any Zd dynamical system has a multiple recurrent point.
Even so, I had been very excited about this topic, it was still surprising for me
to see it explode so spectacularly 10 years later in particularly with the Green-Tao
theorem which heavily relies on such ideas in dynamical systems theory.
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