I explain some general ideal in the theory of diophantine approximation, some of them is original by myself, begin with a toy model, then consider the application on folklore Swirsing-Schmidt conjecture.

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**1. Dirichlet theorem, the toy model **

The very basic theorem in the theory of Diophantine approximation is the well known Dirichlet approximation theorem, the statement is following.

Theorem 1 (Dirichlet theorem)for all is a irrational number, we have infinity rational number such that:

Remark 1It is easy to see the condition of irrational is crucial. There is a best constant version of it, said, instead of , the best constant in the suitable sense for the theorem 1 should be and arrive by at least. The strategy of the proof of the best constant version involve the Frey sequences.

Now we begin to explain the strategies to attack the problem.

\paragraph{Argument 1, boxes principle} We begin with a easiest one, i.e. by the argument of box principle, the box principle is following,

Theorem 2 (Boxes principle)Given and two finite sets , set , if we have a map:

Then there exists a element such that there exist at least two element , .

*Proof:* The proof is trivial.

Now consider, , the sequences , then . Divide in an average way to part: . Then the linear structure involve (which, in fact play a crucial role in the approach). And the key point is to look at and integers.

\paragraph{Argument 2, continue fractional} We know, for irrational number , have a infinite long continue fractional:

And we have,

Then .

\paragraph{Argument 3, Bohr set argument} We begin with some kind of Bohr set:

The key point is the shift of Bohr set, on the vertical line i.e. is very slow, and can be explained by

So:

in But in fact they are not really independent, as the number of Bohr sets increase, then you can calculate the correlation, thanks to the harmonic sires increasing very slowly, wwe can get something non trivial by this argument, but it seems not enough to cover the whole theorem 1.

\paragraph{Argument 4, mountain bootstrap argument} This argument is more clever than 3, although both two arguments try to gain the property we want in 1 from investigate the whole space but not , this argument is more clever.

Now I explain the main argument, it is nothing but sphere packing, with the set of balls

and define its subset

Then , and . If we can proof,

Remark 2If we can proof 3, it is easy to see the theorem 1 follows.

*Proof:* The proof follows very standard in analysis, may be complex analysis? Key point is we start with a ball , whatever it is, this is not important, the important thing is we can take some ball with the center of in , then try to consider to extension and then we find the boudary is also larger then we can extension again, step by step just like mountain bootstrap argument. So we involve in two possible ending,

- The extension process could extension to whole space.
- we can not use the extension argument to extension to the whole space.

If we are in the first situation, then we are safe, there is nothing need proof. If we are in second case, anyway we take a ball . Then try to find good ball to approximate , but this is difficult…

Remark 3Argument 1 is too clever to be true in generalization, argument 2 is standard, by the power of renormalization. argument 3 and argument 4 have gap… I remember I have got a proof similar to argument 4 here many years ago, but I forgot how to get it…

**2. Schimidt conjecture **

The Schimidt conjecture could be look as the generalization of Dirchlet approximation theorem 1 to algebraic number version, to do this, we need define the height of a algebraic number.

Definition 4We say a number is a order algebraic number if and only is the minimal polynomial of , have degree .

Definition 5 (Height)Now we define the height of a th order algebraic number as , Where

Now we state the conjecture:

Theorem 6 (Swiring-Schimidt conjecture)For all transendental number , there is infinitely are th algebraic number such that:

Where is a constant only related to but not .

I point out the conjecture is very related to the map:

Where is the th symmetric sum.

Remark 4is a map , what we consider is its inverse, , but is not smooth, it occur singularity when for some . And the map, as we know, the singularity depend on the quantity .

Remark 5I then say something about the geometric behaviour of the map , as we know, what we have in mind is consider the map as a distortion , Then is just the pullback of the canonical metric on (morally) to .