across 3 years ago I am having trouble understanding the theorem below, so I will post it here as a reminder to myself to review it as often as I can. Also, I may get lucky enough to have someone clarify the questions that I will most likely post here as I read through it time and time again.

1. across

Dirichlet's Approximation Theorem "If $$\alpha$$ is a real number and $$n$$ is a positive integer, then there exist integers $$a$$ and $$b$$ with $$1\leq a\leq n$$ such that $$|a\alpha-b|<1/n$$." Proof. Consider the $$n+1$$ numbers $$0,\{\alpha\},\{2\alpha\}$$, ..., $$\{n\alpha\}$$. These $$n+1$$ numbers are the fractional parts of the numbers $$j\alpha$$, $$j=0,1$$, ..., $$n$$, so that $$0\leq\{j\alpha\}<1$$ for $$j=0,1$$, ..., $$n$$. Each of these $$n+1$$ numbers lies in one of the $$n$$ disjoint intervals $$0\leq x<1/n$$, $$1\leq x<2/n$$, ..., $$(j-1)\leq x<j/n$$, ..., $$(n-1)\leq x<1$$. Because there are $$n+1$$ number under consideration, but only $$n$$ intervals, the pigeonhole principle tells us that at least two of these numbers lie in the same interval. Because each of these intervals has length $$1/n$$ and does not include its right endpoint, we know that the distance between two numbers that lie in the same interval is less than $$1/n$$. It follows that there exist integers $$j$$ and $$k$$ with $$0\leq j < k\leq n$$ such that $$|\{k\alpha\}-\{j\alpha\}|<1/n$$. Now let $$a=k-j$$ and $$b=[k\alpha]-[j\alpha]$$. Because $$0\leq j < k\leq n$$, we see that $$1\leq a\leq n$$. Moreover,$|a\alpha-b|=|(k-j)\alpha-([k\alpha]-[j\alpha])|$$=|(k\alpha-[k\alpha])-(j\alpha-[j\alpha])|$$=|\{k\alpha\}-\{j\alpha\}|<1/n.$Consequently, we have found integers $$a$$ and $$b$$ with $$1\leq a\leq n$$ and $$|a\alpha-b|<1/n$$, as desired. $$\blacksquare$$

2. asnaseer

Your question prompted me to search this topic and try and learn about it. During my google quest, I came across the original German paper on this topic and thought it might interest you as I /think/ you are a German speaker. http://bibliothek.bbaw.de/bibliothek-digital/digitalequellen/schriften/anzeige?band=07-abh/1837&seite:int=00000286

3. No-data

Sabes algo de eso jose?

4. victorarana

si