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With $p$ an integer and $\alpha_i$ a real number, does $\{p\alpha_i\}\leq\epsilon$ hold for some predefined $\epsilon$, maybe depending on the number of $\alpha_i$ values? I looked at the simultaneous version of the Dirichlet's approximation theorem (Wikipedia), but the absolute operator is 'in the way'. Or is such a limit impossible?

Jeroen Boschma
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  • Not sure you got the question right ... I would say ${p\alpha}\leq\epsilon$ where $\alpha$ is a fixed real and $p$ very would make more sense. Otherwise, it is not clear what is the index $i$ for. – Salcio Dec 28 '22 at 14:36
  • @Salcio Question is if there is a limit $\epsilon$ simultaneously valid for all different/arbitrary $\alpha_i$ and a single fixed $p$. The notation follows the Wikipedia text. – Jeroen Boschma Dec 28 '22 at 14:43
  • If $\alpha_1,\alpha_2$ are irrational and $\alpha_1+\alpha_2$ is an integer, you are in trouble. – fedja Dec 28 '22 at 14:55
  • @fedja If ${p\alpha_1}=\epsilon_1$ then ${p(1-\alpha_1)}=1-\epsilon_1$. So maybe you can find a $p$ for which all ${p\alpha_i}$ are (arbitrary) close to $\frac{1}{2}$? – Jeroen Boschma Dec 28 '22 at 15:07
  • @JeroenBoschma That would be a different interpretation of the question from the one I used. So, do I understand it right that you are asking whether For every n, there exists $\varepsilon=\varepsilon(n)\in(0,1)$ such that for every $a_1,\dots,a_n$, there exists $p$ such that ${pa_i}<\varepsilon$ for all $i=1,\dots,n$? – fedja Dec 28 '22 at 17:06
  • @fedja Yes, that's what I mean. One $p$ applied to every $\alpha_i$. From your earlier comment I seemed to understand that this is not possible, so I threw in another option what then would mean $\varepsilon(n)=\frac{1}{2}$. Layman (-person...) overhere... – Jeroen Boschma Dec 28 '22 at 17:25
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    If one fixes $\alpha_1, ...\alpha_k$, such that they are linear independent over $Q$ then the tuples $({n\alpha_1},..., {n\alpha_k})$ should be uniformly distributed in the unit cube $[0,1]^k$. In particular, for any $\epsilon >0$ one can find $n$ such that ${n\alpha_i} <\epsilon$. Check out "Uniformly distributed sequences" by Kuipers – Salcio Dec 28 '22 at 20:49

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